How does a newly imported wallet know which outputs are in the wallet?

There are basically two ways to find out which outputs are in your wallet.

1. They can check block by block since Genesis or the birth of a particular wallet and check if they are the owners of the UTXOs being created and used. This allows you to know not only the current state of your wallet, but also its history.
2. Check the current UTXOset and see if any of the outputs there are owned by the wallet.

My question is, what approach do wallets typically take? I know that many people will adopt the first one because it can load the historical one, but do they also check the UTXO set? Are there any wallets that only check the UTXO set and not the history?