A Crime To Ridicule Police Officers?

Yes, in Hattiesburg, Mississippi:

Sec. 16-1. Abusing policemen.

It shall be unlawful for any person to revile, abuse or ridicule any policeman of the city while in the discharge of his official duties. (Ord. 1504, sec. 1, 5-13-64; Ord. 425, sec. 1--2, 9-17-14)

Pretty clearly unconstitutionally overbroad, especially since I couldn't find any case that limited this to unprotected speech (such as fighting words, which is to say individually targeted insults that are reasonable likely to cause a fight). Even if limited to fighting words, it might still be unconstitutional under R.A.V. v. City of St. Paul. But without such a limitation, the matter is even more clear.

Judging by the archives of the Hattiesburg American, arrests for "ridiculing a police officer" (usually combined with other charges) are not uncommon, though I can't tell from the brief summaries exactly what the "ridicul[e]" consisted of.