There is quite a bit known, although some unknown things (like lower bounds for lowest non-zero eigenvalue) are difficult unsolved problems.
Iwaniec' "Spectral methods in automorphic forms" treats many aspects of your question.
For example, the spectrum of the hyperbolic plane itself is all "continuous", no discrete spectrum.
For general compact surfaces, of course the spectrum is discrete, and Weyl's Law (a theorem...) gives the asymptotics.
For non-compact surfaces, in general it is not clear whether "most" of the spectrum is discrete, or not, since, in general, there is considerable continuous spectrum.
In the case of modular curves, the discrete spectrum is given explicitly in terms of Eisenstein series, and Selberg's trace formula shows that the bulk of the spectrum is discrete. In this setting, a form of Weyl's law is still provable.
The most delicate question seems to be about bounding the lowest non-zero eigenvalue for modular curves, which is Selberg's conjecture.