  Measurement incompatibility has proved to be an important resource for
information-processing tasks. In this work, we analyze various levels of
incompatibility of measurement sets. We provide operational classification of
measurement incompatibility with respect to two elementary classical
operations, viz., coarse-graining of measurement outcomes and convex mixing of
different measurements. We derive analytical criteria for determining when a
set of projective measurements is fully incompatible with respect to
coarse-graining or convex mixing. Robustness against white noise is
investigated for mutually unbiased bases that can sustain full incompatibility.
Furthermore, we propose operational witnesses for different levels of
incompatibility subject to classical operations, using the input-output
statistics of Bell-type experiments as well as experiments in the
prepare-and-measure scenario.
