  We address the use of simple quantum probes for the spectral characterization
of classical noisy environments. In our scheme a qubit interacts with a
classical stochastic field describing environmental noise and is then measured
after a given interaction time in order to estimate the characteristic
parameters of the noise. In particular, we address estimation of the spectral
parameters of two relevant kinds of non-Gaussian noise: random telegraph noise
with Lorentzian spectrum and colored noise with $1/f^{\alpha}$ spectrum. We
discuss the estimation precision achievable by quantum probes by analyzing in
details the behavior of the quantum Fisher information (QFI) as a function of
the noise and the other parameters. We found that population measurement on the
qubit is optimal for noise estimation and also evaluate the optimal interaction
times for the quantum probe, i.e. the values maximizing the QFI and the quantum
signal-to-noise ratio. For random telegraph noise the QFI is inversely
proportional to the square of the switching rate, meaning that the quantum
signal-to-noise ratio is constant and thus the switching rate may be uniformly
estimated with the same precision in its whole range of variation. For colored
noise, the precision achievable in the estimation of ``color'', i.e. of the
exponent $\alpha$, strongly depends on the structure of the environment, i.e.
on the number of fluctuators describing the classical environment. For an
environment modeled by a single random fluctuator estimation is more precise
for pink noise, i.e. for $\alpha=1$, whereas by increasing the number of
fluctuators, the QFI is has two local maxima, with the largest one drifting
towards $\alpha=2$, i.e. brown noise.
