  We provide examples of classical small-cancellation groups which have
non-sigma-compact Morse boundary. These are first known examples of groups with
non-sigma-compact Morse boundary. Some small-cancellation groups do have
sigma-compact Morse boundary, so this property distinguishes quasi-isometry
types of small-cancellation groups. In fact, we give a complete description of
when Morse boundaries of $C'(1/6)$--groups have sigma-compact Morse boundary.
We also provide examples of $C'(1/6)$--groups where all Morse rays are strongly
contracting.
