What does sola scriptura mean?

i) It's become a Catholic trope to say that sola scriptura is self-contradictory, and I've seen Protestant apologists struggle with that charge.

ii) First thing I'd point out is that you can rule out certain options before you know the right answer. You can sense that a proposed answer is wrong before you know what the right answer is. 

iii) The historic target of sola scriptura is the papacy and post-apostolic church councils. Sola scripture is the claim that there are no infallible post-apostolic church councils. Likewise, that God didn't institute the papacy. The pope is not a divine mouthpiece.

iv) Apropos (iii), suppose a Catholic apologist asks us where do we find that in the Bible? But that's the point–we don't find the papacy in the Bible. And we don't find divine promises to inspire post-apostolic church councils in the Bible. We find promises to the apostles. But we don't find comparable promises to bishops or post-apostolic church councils.

v) Catholics sometimes appeal to the "ordination" of Timothy (1 Tim 4:14; 2 Tim 1:6) as an example of holy orders. Suppose, for argument's sake, we agree that the ceremony conferred a "charism" on Timothy. But Paul officiated at that ceremony. So that provides no precedent for "bishops" who aren't handpicked deputies of the apostles. For "bishops" on whom apostles did not lay hands.

vi) Moreover, Paul isn't Peter. At best, the case of Timothy establishes Pauline succession, not Petrine succession. So that's hardly precedent for the papacy. Indeed, that's at odds with the exclusive claims of the papacy. That example is counterproductive to Catholic claims. 

vi) It's not that we don't find sola scriptura in the Bible, in the sense of a direct statement about sola scripture. That's a confused way to frame the issue. Sola scriptura is defined by the point of contrast. The historical alternative. 

We don't find what sola scriptura opposes in the Bible. We don't find a divine mandate or promise regarding the infallible authority of post-apostolic church councils. We don't find a divine promise to protect post-apostolic church councils from error. And we don't find a divine mandate or promise regarding a Petrine succession, where bishops of Rome are oracles of God.

For Catholic apologists to ask or exclaim, "Where do you find that in the Bible?" proves our point. We don't, and that's the problem–for Catholicism. 

vii) Moreover, it's not just an argument from silence. We've seen Roman Catholicism in action. We'd seen how the claimants to special divine guidance perform. We've seen popes and Catholic church councils in action. That doesn't make the Catholic alternative plausible. To the contrary, that brings the Catholic alternative into disrepute.

viii) Catholics appeal to Acts 15, but apostles along with a sibling of Jesus were participants. So that's no precedent for post-apostolic church councils. 

ix) Moreover, Catholics distinguish between local councils and ecumenical councils. But we don't find that distinction in Scripture. There's no divine promise regarding ecumenical councils in contrast to local councils.

x) Sometimes the debate is framed in terms of cessationism v. continuationism, but that separable. It's true that cessationism rules out the kind of divine guidance and protection from error that Rome lays claim to, but so does continuationism inasmuch as continuationism, if true, is not a promise or expectation directed at the papacy or Roman bishops in council. Rather, continuationism, if true, applies to the laity in general. 

xi) There are, of course, Catholic prooftexts regarding the papacy, tradition, apostolic succession, and "the Church", but that's different from the allegation that sola scriptura is self-contradictory. And Protestants regard the Catholic prooftexts as bogus. 

xii) Catholic apologists might objection that it begs the question to say Catholic prooftexts are bogus. "By what authority" do we make that value judgment? Yet it must be possible to assess Catholicism independent of Catholicism–otherwise, nobody would ever be in a position to convert to Catholicism. Likewise, it must be possible for cradle Catholics to assess Catholicism independent of Catholicism. After all, the fact that you were born into a particular religious (or irreligious) tradition carries no guarantee or even presumption that you were born into the "right" religious tradition–inasmuch as people are born into different, competing religious (or irreligious) traditions.