According to Sola Scriptura, the Bible alone is the ultimate and only infallible authority (in terms of special revelation). Here is a good statement of the position from the Westminster Confession of Faith:
The supreme judge by which all controversies of religion are to be determined, and all decrees of councils, opinions of ancient writers, doctrines of men, and private spirits, are to be examined; and in whose sentence we are to rest; can be no other but the Holy Spirit speaking in the Scripture. (WCF 1:10)
Most advocates of Sola Scriptura assert that we ought to listen very carefully to the historic teachers and councils of the Church, and that we ought to default to trust in their judgment much of the time. If we find our own interpretations of Scripture to be contrary to theirs, this should give us a weighty pause.
But here's the real question: What happens when we've listened very carefully to the opinions of the Church Fathers, the historic teachers of the Church, Church councils, popes, etc., and still it seems to us, upon as careful analysis as possible, that what they have said contradicts what the Bible says? What do we do then? The primary founder of the idea of Sola Scriptura (Martin Luther, of course) had no doubt on this point:
Unless I am convinced by Scripture and plain reason - I do not accept the authority of the popes and councils, for they have contradicted each other - my conscience is captive to the Word of God. I cannot and I will not recant anything for to go against conscience is neither right nor safe. God help me. Amen.
If our interpretation of Scripture contradicts the unanimous consensus of the historic Church, that should give us great pause. It should cause us to seriously and very carefully reconsider our position. But, in the end, according to Sola Scriptura, it should not cause us to give up our position, because ultimately only the Bible is infallible.
This is obviously the logical implication of Sola Scriptura. However, I have found that when one articulates this too plainly and clearly, one often finds oneself accused of misrepresenting Sola Scriptura. "That's not what Sola Scriptura means! You're twisting it! Sola Scriptura doesn't mean we should just go off by ourselves with our Bibles and ignore the whole history of the Church, all the ancient teachers, etc.! On the contrary, a right understanding of Sola Scriptura tells us to treat with great deference and respect the traditions of the Church and to do our interpreting in the context of those who have gone before, and to be taught by the Fathers, the councils, and the great creeds and confessions of the historic Church!"
Notice that this response completely misses the point. I never said that Sola Scriptura means that we should go off by ourselves and read the Bible without input from anyone else, or that we shouldn't have great respect and deference for the traditions of the Church, etc. All I said was that, when push comes to shove, after I've listened to all the input I can get, and after laborious and careful examinations and re-examinations--if, after all of that, it still appears to me that the Bible disagrees with the unanimous consensus of the historic Church, Sola Scriptura tells us to go with our own interpretations of Scripture over the consensus of the historic Church.
I think that putting this so plainly makes a lot of Protestants very uncomfortable, especially those (like Presbyterians, Anglicans, etc.) who want to distance themselves from all those "hillbilly" evangelicals who go off with "just the Bible and me," no education, no awareness of Church history, etc., and come up with goofy personal interpretations and then start their own new church in their living room. Perhaps subconsciously to avoid this discomfort, they try to make distinctions that aren't there, such as the distinction Protestant teacher Keith Mathison has made between Solo Scriptura and Sola Scriptura, where the former is the hillbilly evangelical version and the latter is the historic, tradition-minded version. Unfortunately, as many have pointed out, no principled distinction really exists to be made. Sure, various people who try to practice Sola Scriptura are more or less historically-informed, literate, respectful of traditional positions, etc., but the basic principle is the same--the Bible alone is infallible and so the Bible should be followed even if it should seem to contradict everyone else. I'm not aware of any serious position adopted by any Protestant church which says that we should read the Bible while ignoring all input from anyone else or from earlier teachers in the Church. Every Protestant group that I am aware of that has articulated any clear position on this affirms that tradition is valuable, that we should know history, etc. They simply add that none of those things are infallible, and the Bible alone is. And that is the same position that is advocated by Keith Mathison and all the cultured Presbyterians and Anglicans as well.
Those who say that Sola Scriptura is being misrepresented when it is said that it implies that we must follow our own interpretations of the Bible even, if need be, against the unanimous position of the historic Christian Church need to think more carefully through their own doctrine. If we are not to follow our own interpretations as the ultimate authority, what is the alternative? Are we to abandon what it seems to us that the Bible is saying in order to defer to historic councils, teachers, etc.? This makes no sense in the context of Sola Scriptura, for it would be to treat these ancient councils and teachers as if they were infallible, and it would be to give up the practice of checking what they said against the Scriptures before believing them, which is the whole point of Sola Scriptura! Imagine if Luther had done this:
Unless I am convinced by Scripture and plain reason - I do not accept the authority of the popes and councils, for they have contradicted each other - my conscience is captive to the Word of God . . . Well, on second thought, if all those popes and councils and teachers disagree with me, I should probably just trust them, even if to me it looks like the Bible contradicts them. Who am I to follow my own interpretations of Scripture over against the historic positions of the Church? Never mind, then. I recant.
That would have been the end of the Protestant Reformation!
Sola Scriptura tells us that we are not to trust implicitly any authority outside of Scripture--whether that be ancient fathers, councils, popes, large groups of Christians, or whatever. We are only to trust the Bible implicitly, because only it has infallible authority. This implies, as the Westminster Confession put it, that we must test or examine all the ideas of men--ancient or modern, majority or minority--in the light of Scripture. But Scripture must be interpreted. Whose interpretation, then, are we to finally trust? Of course, it has to be our own. If we were to trust implicitly the interpretations of ancient writers, that would be the same as to give up testing their opinions by any standard higher than themselves. If we must test all these opinions by the Bible alone, the only thing that that can mean is that we must try as hard as we can to figure out what we think the Bible is saying, and then judge what everyone else is saying--including what everyone else says the Bible is saying--by our own interpretations of the Bible. This is not a misrepresentation of Sola Scriptura; this is Sola Scriptura!
And yet, as I mentioned, those who try to point this out clearly and plainly typically get accused of misrepresenting or over-simplifying Sola Scriptura. And those who actually try consistently to practice this have an even worse time! I can attest to this personally. As I discuss in greater detail here, back when I was a Presbyterian (in the Orthodox Presbyterian denomination), I came to certain convictions about church government that I believed to be biblical (and historically Presbyterian as well). These convictions (along with some others) led me to the conclusion that the Orthodox Presbyterian Church (OPC) is in schism, and that ideally we all ought to join another Presbyterian denomination. The response of the pastor of my OPC was, partly, to accuse me of arrogance for asserting my own opinions on these matters against the dominant judgment of the Reformed tradition (as he thought). It didn't matter that I claimed to believe my position on biblical grounds; the mere fact that I would disagree with so many others was enough to prove how presumptuous and conceited I was. He also accused me of intentionally defying the authority of the Church. This reminded me of Martin Luther's trial, and I told him so. He was not impressed. He told me I was misrepresenting Sola Scriptura and misusing it to avoid having to listen to the Church. So it appears that it is one thing to proclaim Sola Scriptura and the authority of the Bible over "popes and councils" in the abstract, but when one tries to apply this to some particular leader or church tribunal, one finds that Sola Scriptura had better not be brought out of the abstract and into conflict with the currently prevailing tradition!
Jimmy Akin articulates this situation well:
There may, in any given congregation, be a number of theologically inclined people who make a serious study of the Bible, but the average person simply listens to the exposition given to the Scriptures in the Sunday sermon or weekday Bible study and accepts it. The pastor or Bible study leader sketches out what his view of the Scripture is, and it is not rigorously questioned by the average listener. The average person does not go out and get commentaries from opposing viewpoints, compare them to the pastor’s view, and then do a rigorous analysis of the arguments on both sides. . . .
When it is exercised consistently, when the individual really does give his pastor’s and denomination’s teaching a rigorous analysis, he is likely to find out that his previous theological sloth is not the only barrier to his ability to exercise the absolute right of private judgment. There is a second barrier, for if he comes to the conclusion that the pastor or the denomination is simply wrong about something it considers important, then he is almost always confronted with two choices: Keep your mouth shut about this and don’t go advocating your private interpretation in the congregation (that would prompt a crisis of leadership and disturb the tranquillity of the sheep) or simply get out and take your private interpretation somewhere else.
Any layman trying to stay in a congregation and advocate a different position on something the pastor or the denomination considers important will have first subtle and then no so subtle pressure put on him to either keep quiet or leave, and if he will do neither then he will finally be expelled from the body. . . .
The hypocritical thing is not the expulsion of dissenters, but the holding out of the promise of private interpretation, of promoting it by continual rhetorical harping on this theme and by encouraging the faithful to look down their noses at denominations which don’t preach this principle, when in reality the promise is: “You have an absolute right to interpret the Scriptures for yourself, but we will cast you out if you disagree with something we consider important.”
Most Protestant traditions, then, tend to use Sola Scriptura selectively, as something to be turned on at the proper times and turned off at other times, depending on whether it will be helpful or more trouble than it's worth. If you don't like "Roman innovations" or other historic traditions of the Church, and you want to challenge them based on your own interpretations of Scripture, that is the time to turn it on. But if someone points out how subjective it is, or uses it to challenge your own traditions, then that is the time to turn it off and accuse your opponent of misrepresenting it. (I'm not suggesting that any of this is done consciously, but it is often done nonetheless. And, of course, all of this is assuming that Scripture alone is insufficiently clear to determine many important doctrinal disputes, which I believe to be the case.)
I will end with a selection from St Francis de Sales, who, as is so often the case, has articulated some of the key issues in a way that really hits home. He describes well what Sola Scriptura, applied consistently and thoroughly, really looks like in practice, and how Protestant teachers so often allow it to a point only to stifle it arbitrarily when it starts to get out of hand. This is from The Catholic Controversy (Rockford, IL: TAN Books and Publishers, 1989), 228-230 (I've adjusted the spacing to fit better in blog format):
But what need have I to go far? Beza says in the Epistle to the King of France, that your reform will refuse the authority of no Council; so far he speaks well, but what follows spoils all: "provided," says he, "that the Word of God test it."
But, for God's sake, when will they cease darkening the question! The Councils, after the fullest consultation, when the test has been made by the holy touchstone of the Word of God, decide and define some article. If after all this another test has to be tried before their determination is received, will not another also be wanted? Who will not want to apply his test, and whenever will the matter be settled? After the test has been applied by the Council, Beza and his disciples want to try again? And who shall stop another from asking as much, in order to see if the Council's test has been properly applied? And why not a third to know if the second is faithful?--and then a fourth, to test the third? Everything must be done over again, and posterity will never trust antiquity but will go ever turning upside down the holiest articles of the faith in the wheel of their understandings.
We are not hesitating as to whether we should receive a doctrine at haphazard, or should test it by the application of God's Word. But what we say is that when a Council has applied this test, our brains have not now to revise but to believe. Once let the canons of Councils be submitted to the test of private individuals,--as many persons, so many tastes, so many opinions.
The article of the real presence of Our Lord in the most Holy Sacrament had been received under the test of many Councils. Luther wished to make another trial, Zwingle another trial on that of Luther, Brentius another on these, Calvin another,--as many tests so many opinions. But, I beseech you, if the test as applied by a General Council be not enough to settle the minds of men, how shall the authority of some nobody be able to do it? That is too great an ambition.
Some of the most learned ministers of Lausanne, these late years, Scripture and analogy of faith in hand, oppose the doctrine of Calvin concerning justification. To bear the attack of their arguments no new reasons appear, though some wretched little tracts, insipid and void of doctrine, are set a-going. How are these men treated? They are persecuted, driven away, threatened. Why is this? "Because they teach a doctrine contrary to the profession of faith of our Church." Gracious heavens! the doctrine of the Council of Nice, after an approbation of thirteen hundred years, is to be submitted to the tests of Luther, Calvin, and Beza, and there shall be no trial made of the Calvinistic doctrine, quite new, entirely doubtful, patched up and inconsistent! Why, at least, may not each one try it for himself? If that of Nice has not been able to quiet your brains, why would you, by your statements, impose quiet on the brains of your companions, who are as good as you, as wise and as consistent? Behold the iniquitousness of these judges; to give liberty to their own opinions they lower the ancient Councils, while with their own opinions they would bridle those of others. They seek their own glory, be sure of that; and just as much as they take away from the Ancients do they attribute to themselves.
For more on Sola Scriptura, see here and here. For more on how Anglicans in particular tend to want to "have their cake and eat it too" when it comes to Sola Scriptura, see here and here. For a great article responding to Keith Mathison's attempt to defend Sola Scriptura from the subjectivity of ultimate reliance on individual private interpretation, see here. For a shorter version of the same article, see here. For another great selection from St. Francis de Sales on the ineffectiveness of Sola Scriptura to solve doctrinal disputes, see here.