  We derive fundamental bounds on the maximal achievable precision in
multiparameter noisy quantum channel estimation, valid under the most general
entanglement-assisted adaptive strategy, which are tighter than the bounds
obtained by a direct use of single-parameter bounds. This allows us to study
the issue of the optimal probe incompatibility in the simultaneous estimation
of multiple parameters in generic noisy channels, while so far the issue has
been studied mostly in effectively noiseless scenarios (where the Heisenberg
scaling is possible). We apply our results to the estimation of both unitary
and noise parameters, and indicate models where the fundamental probe
incompatibility is present. In particular, we show that in lossy multiple arm
interferometry the probe incompatibility is as strong as in the noiseless
scenario. Finally, going beyond the multiple-parameter estimation paradigm, we
introduce the concept of \emph{random quantum sensing} and show how the tools
developed may be applied to multiple channels discrimination problems. As an
illustration, we provide a simple proof of the loss of the quadratic advantage
of time-continuous Grover algorithm in presence of dephasing or erasure noise.
