Does your signature have to be cursive?

Answer 1

Yes. In a letter to a close friend you may not have to, but as for formal letters, always sign cursive.

...that's not true. US law states "A signature may be written by hand, printed, stamped, typewritten, engraved, photographed, or cut from one instrument and attached to another, and a signature lithographed on an instrument by a party is sufficient for the purpose of signing it, it being immaterial with what kind of instrument a signature is made. ... whatever mark, symbol, or device one may choose to employ as a representative of himself is sufficient ... The name or mark of a person, written by that person at his or her direction. In commercial law, any name, word, or mark used with the intention to authenticate a writing constitutes a signature. UCC 1-201(39), 3-401(2). A signature is made by use of any name, including any trade or assumed name, upon an instrument, or by any word or mark used in lieu of a written signature."

...so a signature can technically be whatever you want it to be.