This is an excellent question, I think.
Topological spaces could be crudely --- and I mean crudely --- divided into two kinds:
The geometric. These are the spaces that come up all over geometry and algebraic topology — manifolds, CW complexes, configuration spaces, CW complexes, etc. These are almost invariably Hausdorff, though there are plenty of compact Hausdorff spaces that are often thought of as "pathological", such as the Cantor set. I don't much like terms such as "nice space" and "pathological", because although they might be intended harmlessly, they sound to me a bit dismissive towards the second kind of space.
The spectra. I'm using this term loosely (and not in the sense of homotopy theory), but I mean things like the spectrum of a ring, the spectrum of a Boolean algebra (= a compact Hausdorff totally disconnected space), the maximal spectrum of a C*-algebra, and then things such as Julia sets, dynamical attractors, and solutions to iterated function systems, all of which have a spectrummy feel to me.
The spectra seem a bit unloved. No one denies the importance of, say, Spec of a commutative ring, but still, I reckon that most mathematicians subconsciously regard geometric spaces as the primary kind, and sometimes the spectra simply get swept away as "pathological".
Edit: Don't read this without also reading Ilya Grigoriev's comment below!