TLDR: In the US, FinCEN, the Financial Crimes Enforcement Network, introduced guidance for the crypto industry covering the Travel Rule on May 9, 2019. It states that the Travel Rule has been in effect for crypto businesses since 2013. It requires VASPs to implement the travel rule for transaction amounts above $3000 and does not explicitly cover any requirements for non-custodial wallets.
The US Travel Rule, which has been part of the Bank Secrecy Act (BSA) for many years, provides the basis for the FATF Wire Transfer rule. As such, the US regulator FinCEN has insisted that this was applicable ever since they issued their original 2013 guidance on Virtual Currency Exchanges.
One thing to note is that FinCEN mentions the current US BSA rule, under which a transfer of $3,000 or more triggers the Travel Rule. It is unsure if they will implement the FATF recommendation of $1,000 in the future.
It is also unclear if FinCEN will require ownership proofs for non-custodial wallets as part of the Travel Rule. In their 2019 guidance, FinCEN explicitly states that non-custodial wallets are not considered money transmitters - and thus the Travel Rule does not apply to them.