I am Catholic and I don't believe Sola Scriptura. But here I want to engage in some friendly theologizing, trying to figure out what would be the best thing for me to say about Sola Scriptura were I evangelical. The main difficulty for Sola Scriptura is the standard self-defeat argument. Evangelicals typically take Sola Scriptura to be an important Christian doctrine, important enough that one can base theological arguments on it (e.g., arguing against some Catholic or Orthodox belief on the grounds that that belief is not found in Scripture). But let us take Sola Scriptura to be the claim that all true, Christian doctrines are found explicitly or implicitly in Scripture. Then, we have a self-defeat argument against Sola Scriptura: it is proposed as a true, Christian doctrine, but it is nowhere found explicitly or implicitly in Scripture, and hence by its own claim is not a true, Christian doctrine.
The fact that Sola Scriptura is not found in Scripture might be disputed. A standard proof-text for Sola Scriptura is 2 Timothy 3:16-17 which says that Scripture is inspired by God and has as its purpose that one might be "thoroughly equipped for every good work" (NIV; one may also query points in the translation). But of course the opponent of Sola Scriptura does not need to deny that all Scripture is inspired by God. Moreover, the claim that Scripture exists to equip us for every good work does not entail that Scripture is all that is needed to equip us thoroughly for every good work. After all, plainly, lots of other things are needed—air, food, water, intellectual skills, and, above all, God's grace. And even if Scripture were sufficient to equip us for every good work, it would not follow that Scripture contains all true, Christian doctrine. Finally, it is very unlikely that 2 Timothy 3:16-17 contains Sola Scriptura, since the "Scriptures" referred to are the ones Timothy learned "from infancy" (v. 15), and hence are the Old Testament. And the Old Testament surely does not contain all true, Christian doctrines. In fact, when this text was penned, Scripture was not yet completed, and there were surely Christian doctrines not yet in Scripture (such as the Christian doctrines taught in the next chapter of 2 Timothy!).
Nor is it likely that Sola Scriptura would be found in Scripture, since at the points at which most of the New Testament was being written, there was much reliance on apostolic preaching, or on reports of apostolic preaching.
So, what can an evangelical say in defense of Sola Scriptura given the self-defeat argument? One suggestion is to limit the scope of what is claimed. Thus, instead of claiming that Scripture contains all Christian doctrine, one instead claims that Scripture contains all the Christian doctrine that is necessary for salvation. A problem with this more limited claim is that it makes Sola Scriptura a not very interesting doctrine on standard evangelical views of what is necessary for salvation, namely faith that Jesus Christ is Lord. On such views, one can seemingly replace the claim that Scripture is sufficient for salvation with the stronger claim that some collection of three or four verses is sufficient for salvation. And surely one doesn't want Sola Scriptura to simply follow from the sufficiency of three or four verses.
I want to suggest that a better answer to the self-defeat argument is to say that the argument does not show that Sola Scriptura is false. Rather, the self-defeat argument only shows that Sola Scriptura is not a true, Christian doctrine, i.e., that it is either not true, or not a Christian doctrine, or neither. The evangelical can opt for saying that while Sola Scriptura is true, it is not a Christian doctrine. After all, many true claims, even claims about Scripture, are not Christian doctrine. For instance, it is true that Scripture has been translated into Swahili, or that most Bibles are printed in mostly black ink, but these facts are not Christian doctrines. This solution is not original to me—I heard it from a Protestant friend, I think.
Now this way of taking Sola Scriptura has a pleasant ecumenical consequence. It is not appropriate for an evangelical to consider a Catholic or Orthodox Christian to be unorthodox for denying Sola Scriptura. For only the denial of a Christian doctrine can make a Christian unorthodox, and Sola Scriptura is not a Christian doctrine. This reduces the division between evangelicals and Catholics and the Orthodox, though division remains on the other side (Catholics believe that the denial of Sola Scripture is a true, Christian doctrine, and there is no parallel self-defeat argument against their belief here).
Moreover, one might query the epistemological basis of affirming Sola Scriptura once one no longer takes it to be a Christian doctrine. After all, if it is not a Christian doctrine, then one cannot know it one the basis of public divine revelation. One might claim to believe Sola Scriptura on the basis of a private revelation (an angel whispering the doctrine to one), but that is unlikely to convince many others. Could one, perhaps, know Sola Scriptura empirically or maybe by a careful application of a priori reason? I doubt it. Surely one cannot know it empirically. Nor does it seem at all a candidate for a priori knowledge. Maybe one might think there is some way to combine empirical and a priori reasoning with divine revelation to get Sola Scriptura, but I doubt this.
If Sola Scriptura is not a matter of faith (since it's not a Christian doctrine), and cannot be known to be true, I think what would be most reasonable for an evangelical, short of chucking Sola Scriptura altogether, would be to take Sola Scriptura to either be a negative first person claim—"I am not aware of any source of true, Christian doctrine other than Scripture"—or as a working hypothesis.
What is interesting is that in both cases there should be an in-principle openness to the possibility of other loci of divine revelation, such as the Tradition that Catholics and the Orthodox refer to. Adopting either the "negative first person claim" or the "working hypothesis" view of Sola Scriptura would, thus, move ecumenical dialog forward. One might, of course, think this is a minus, but I don't.