My husband found one swimming in our pool.  He couldn't catch it and the rat made it's way under our fence into the neighbor's yard.

Question about trap setting- what do you do with a rat once you catch one?

First, I hope you added some extra chlorine to your pool.   Rats can harbor diseases,  I would hate to think I was swimming where one had been.

I only use "kill" traps and I place the dead rat in a large plastic zip-lock type bag and then wrap that in a trash bag and dispose of it in the garbage can.  

I don't use live traps for the reason you asked,  "what do you do with a live rat ??"

Orange Zone neighbor

When I caught the rat in a live trap, I left it in the shed where it was caught until it died.  A friend who caught one had a barrel of water handy, dumped the rat in the barrel and closed the top until it was no longer with us.

I dont like rats anymore than anyone else but starvation/ drowning them seems cruel. 

Spend a few dollars over at Home Depot or Lowe's and buy the old time rat traps.

One "SNAP" and its over.  They are reliable and reusable.

I bait mine with peanut butter and rinse them in a bucket of bleach water each time I have to remove a dead rat.  Luckily I have not seen or caught any in the last week.