  In quantum mechanics performing a measurement is an invasive process which
generally disturbs the system. Due to this phenomenon, there exist incompatible
quantum measurements, i.e., measurements that cannot be simultaneously
performed on a single copy of the system. It is then natural to ask what the
most incompatible quantum measurements are. To answer this question, several
measures have been proposed to quantify how incompatible a set of measurements
is, however their properties are not well-understood. In this work, we develop
a general framework that encompasses all the commonly used measures of
incompatibility based on robustness to noise. Moreover, we propose several
conditions that a measure of incompatibility should satisfy, and investigate
whether the existing measures comply with them. We find that some of the widely
used measures do not fulfil these basic requirements. We also show that when
looking for the most incompatible pairs of measurements, we obtain different
answers depending on the exact measure. For one of the measures, we
analytically prove that projective measurements onto two mutually unbiased
bases are among the most incompatible pairs in every dimension. However, for
some of the remaining measures we find that some peculiar measurements turn out
to be even more incompatible.
