If the singularity is spacelike (as in a Schwarzschild black hole) then the answer is no, simply because the singularity is in the future. Information about the singularity would only be available in the future light cone of the singularity, and there is none because the singularity is the end of time. You won't see the destruction of the camera because it is spacelike separated from the destruction of you.
For a question like this, where only causality matters and not tidal forces, you don't need to worry about curved spacetime or general relativity at all: there are special-relativistic analogues of the situation that preserve all of the important features. My answer to that other questionMy answer to that other question covers this with a setup involving synchronized time bombs.
If the singularity is timelike, as in the charged and/or rotating vacuum solutions, then it can be detected and studied in the same way as any ordinary object. (But not by taking selfies.) It can also be avoided like any ordinary object. These solutions probably aren't realistic, though, since they are also (if you avoid the singularity) traversable wormholes.
