I'm sure this isn't a popular opinion but I would use a catch and release trap for a mouse or rat. It really bothers me to kill any creature.
When my husband had to clean out his parents house after his mother passed away he found out that a small room that his father had built on to the house and used as a storage room had a mother raccoon and babies inside. His father had passed away many years earlier and his mother had never gone into the room. Somehow momma raccoon had found a way to get in and felt it was safe for her babies. My husband called an animal control company to remove them. We both assumed that they would trap them and release them into the many wooded areas surrounding the house.
Fortunately the day before they were to come to the house we found out that they were just going to kill both mom and babies. That was totally unacceptable to us. What we did was turn on loud music in the shed and leave the door open. The next morning they were gone. We watched on the cameras and saw mom take each baby in her mouth and one by one she carried them across the street and into the woods. She had five babies altogether and it was so heartwarming to see how she cared for them.
Even though I don't care for raccoons because I'm slightly afraid of them, there was no way that I could have ever allowed them to be killed like that.
The company had given my husband a business card with a picture of a raccoon on it so a few days after mom and babies moved away he took the card and handed it to me. On it he had written the words coming from the raccoon's mouth "thank you for saving my life". I still have the card sitting on our desk. I smile everytime I look at it.