The transaction pool is not stored on the blockchain: it is the set of transactions that are used to feed new blocks. Nodes store it in memory, and a
poolstate.bin file when the node exits.
The wallet currently scans the txpool for new transactions. If you run
show_transfers pool in
monero-wallet-cli, you will see them, for example. I believe
monero-wallet-gui also shows them, but I'm less sure about it.
A useful, but mostly unknown feature in
monero-wallet-gui is incoming transaction tracking: in the Receive page, there is a "Tracking" line which will check the blockchain and txpool in realtime for transactions matching the settings (payment ID and optional amount) set above it. If you are on this screen while waiting for your transaction to arrive, you will see that line change as your transaction is seen in the txpool, then in the blockchain.
I'll quote the "Help" link's text here for more information:
This is a simple sales tracker:
Click Generate to create a random payment id for a new customer
Let your customer scan that QR code to make a payment (if that customer has software which supports QR code scanning).
This page will automatically scan the blockchain and the tx pool for incoming transactions using this QR code. If you input an amount, it will also check that incoming transactions total up to that amount.
It's up to you whether to accept unconfirmed transactions or not. It is likely they'll be confirmed in short order, but there is still a possibility they might not, so for larger values you may want to wait for one or more confirmation(s).