signal of $\frac{1}{f}$ Noise
Power-law behaviors in frequency can be found in several unrelated observations and systems. This is apparently the case for $1/f$ or flicker noise. Note that an exact $\alpha=1$ exponent might be too stringent, and people can be interested in a wider range, like  $1/f^\alpha$ with $1/2\le\alpha\le3/2$.

The plot above is from 1/f noise.
Since only looking at the decay of a PSD does not provide you with much ideas about the morphology of signals, you might be interested in looking at generative processes, or accumulated actions that may generate such types of spectra, from physics phenomena (superposition of relaxation processes, infinitely large fluctuations, noise in diffusion processes etc.). And then maybe find analogies when the phenomenon you are studying.
Here are three papers of interest:

$1/f$  noise: a pedagogical review by Edoardo Milotti
$1/f$ noise’’ in music: Music from $1/f$ noise by Richard F. Voss and John Clarke


The spectral density of fluctuations in the audio power of many
musical selections and of English speech varies approximately as $1/f$
(f is the frequency) down to a frequency of 5×10−4 Hz. This result
implies that the audio‐power fluctuations are correlated over all
times in the same manner as ’’$1/f$ noise’’ in electronic components.
The frequency fluctuations of music also have a $1/f$ spectral density
at frequencies down to the inverse of the length of the piece of
music. The frequency fluctuations of English speech have a quite
different behavior, with a single characteristic time of about 0.1 s,
the average length of a syllable. The observations on music suggest
that $1/f$ noise is a good choice for stochastic composition.
Compositions in which the frequency and duration of each note were
determined by $1/f$ noise sources sounded pleasing. Those generated by
white-noise sources sounded too random, while those generated by
$1/f^2$ noise sounded too correlated.


Power laws, Pareto distributions and Zipf's law