How are Sunday School teachers chosen in your church?

We choose to ask teachers from our attenders. They do not have to be members, but they must sign an accountability document and be screened for sexual abuse prevention.

Our elders work with the Sunday School Superintendent, either suggesting, confirming, or rejecting teachers.

We look for godly individuals who know the Lord, embrace sound doctrine, live a respectable life, and are faithful.

I voted "established vetting..."
There are qualifications specified in our constitution. These are not the same as pastor/deacon qualifications, however. We do expect them to have some evident giftedness. In a church our size, that's not usually hard to determine.
... and interest in serving in this way tends to correspond to ability, though there are exceptions to that.
(We also screen for criminal history, mainly with sexual abuse history as the concern there)

Ed... like Angela, I'm curious why your teachers do not have to be members. How would you handle church discipline, for example? Perhaps the accountability statement covers that but would this statement really differ much from, say, a church covenant?

None of the choices in the poll accurately represent what we do. We have two SS superintendents, one for adult classes and one for children's classes. These men observe church members and approach them about teaching if they believe them to be spiritually mature, the right kind of examples, etc. (This goes to what you said about a pattern of behavior.) For a youth class teacher, if they accept, they also have to have a background check, as we do for all adults who work with children in any capacity, nursery, etc.

We have come to the conclusion that we need to have a more rigorous process to determine those who are truly "apt to teach," but we haven't worked through that whole process yet. We do have ongoing teacher training courses, one of which we are going through right now. We don't want to just throw warm bodies at SS classes, but we also don't want to offend those believers who are gung-ho about ministry and getting involved, but wouldn't necessarily make good teachers (at least not yet). Sometimes, we use some of these as teacher's helpers, so they can learn. We have to balance that against not having enough qualified people and then having to combine various classes. I guess what I'm saying is that we don't entirely have it figured out, but we are continually working on the process.

For curriculum, we use different things for the adults. The next one we will be doing is an intense "new members" class. This is something we are improving for the new members, but we also want to have a refresher for the adults who are long-time members to remind them of the duties, responsibilities and benefits of church membership, and what that really means to the growth of the body of Christ.

For kids, our church generally uses RBP, but sometimes we do something different. I teach in the 4th-6th grade classes, working together with a doctoral seminary student, and this past semester, we thought we would try something a little more intense than the usual stuff RBP does at this level. We did a series on the teachings and parables of Christ that we worked out on our own, rather than the usual "Life of Christ." Took more time and research but was very rewarding.