Does crossing over occur in sex chromosomes, X and Y?

Excellent question

They do cross over in certain regions called pseudoautosomal regions.

The pseudoautosomal regions, PAR1, PAR2, and PAR3, are homologous sequences of nucleotides on the X and Y chromosomes. Although genetic recombination (occurring during sexual reproduction) is known to be limited only to the pseudoautosomal regions (PAR1 and PAR2) of the X and Y chromosomes, a 2013 study reports allelic unequal recombination between the X-chromosome-transposed region (XTR) Yp11.2 and Xq21.3, indicating the presence of a new PAR, which has been named PAR3. This PAR3 region is exhibited in 2% of the general population. Also, the additional layer of justification has been provided from another study on six dyslexic cases[4] which were shown to harbor duplications and deletions in the same Xq21.3 and Yp11.2 regions through allelic unequal recombination.