Small-cancellation groups with and without sigma-compact Morse boundary

We provide examples of classical C'(1/6) -small-cancellation groups which have non- \sigma -compact Morse boundary. These are first known examples of groups with non- \sigma -compact Morse boundary. Some C'(1/6) -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.