Other sources are consistent with what is here -- that the adverb carries the "s", the adjective doesn't, and that English English speakers lean toward the "s", while American English speakers drop it, for the most part. I, however, tend to use the "s" as an adverb, and I'm American.

"In regard to" and "with regard to" should never carry the "s". You may send a note or gift "with best regards", but refer to something "with regard" to whatever. The added "s" when using "regard" while referring to something is a pet peeve.

Ditto with people -- lots of people, many of them well-educated -- who say "eco cetera" instead of "et cetera." Drives me nuts. As does "mis-chee-vee-us" (vs mis-chi-vus), nu-cu-lar (vs nu-klee-ar), jew-ler-y (vs jew-el-ry) all annoy me.

Though there are greater crimes on the planet than annoying me...