The Statute of Frauds sets the limit over which a contract has to be in writing, and other things. Not all contracts have to be in writing. Additionally, while contracts are generally signed by both parties, it actually, when a written/signed contract is required, only needs to be signed by the party against whom a remedy is being sought.

If agree to sell me your bicycle in a written contract for $1,500 and you sign it and I pay, that contract doesn't need my signature for me to seek a legal remedy against you in the event you don't deliver the bicycle.

Only under certain scenarios do contracts have to be written, never mind signed. Very often, handshake or verbal contracts are perfectly enforceable.