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.
