I know (and this is of course rather elementary) that an isolated point in the spectrum of a self-adjoint operator $T$ always belongs to the point-spectrum.

I would like to ask: Are there similar characterizations for singular continuous and absolutely continuous spectrum as well?

In fact, I believe that there should be rather characterizations of sets which cannot support s.c. and a.c. spectrum?- Is that true?

Is it possible to have for example a.c. spectrum on an isolated set of measure zero? Or are there further restrictions on this set so that such a set can no longer support this type of spectrum?