  Different approaches to quantum gravity generally predict that the dimension
of spacetime at the fundamental level is not 4. The principal tool to measure
how the dimension changes between the IR and UV scales of the theory is the
spectral dimension. On the other hand, the noncommutative-geometric perspective
suggests that quantum spacetimes ought to be characterised by a discrete
complex set -- the dimension spectrum. Here we show that these two notions
complement each other and the dimension spectrum is very useful in unravelling
the UV behaviour of the spectral dimension. We perform an extended analysis
highlighting the trouble spots and illustrate the general results with two
concrete examples: the quantum sphere and the $\kappa$-Minkowski spacetime, for
a few different Laplacians. In particular, we find out that the spectral
dimensions of the former exhibit log-periodic oscillations, the amplitude of
which decays rapidly as the deformation parameter tends to the classical value.
In contrast, no such oscillations occur for either of the three considered
Laplacians on the $\kappa$-Minkowski spacetime.
