  We study generic fractal properties of bounded self-adjoint operators through
lower and upper generalized fractal dimensions of their spectral measures. Two
groups of results are presented. Firstly, it is shown that the set of vectors
whose associated spectral measures have lower (upper) generalized fractal
dimension equal to zero (one) for every $q>1$ ($0<q<1$) is either empty or
generic. The second one gives sufficient conditions, for separable regular
spaces of operators, for the presence of generic extreme dimensional values; in
this context, we have a new proof of the celebrated Wonderland Theorem.
