What is the most humane way to kill mice ?

Get a cat

I second a cat. It might just drive them away without even killing them. They hate the smell of cats.

Buying a cat means spending money. Just train a street cat with some cat food and entice him to get in the shed to scare them all away!!

I think if you put food in the shed and leave the door open for “a street cat” then the mice will be eating that food, as well as other animals that will come for the free meal. I think if you get a regular, household pet cat, it’s not going to want to stay in the shed to kill the mice but it may sniff around in there and scare them. I’m not for rat poison but it may be the only way to go, or an exterminator. Also, eliminate any food sources you have stored, like birdseed. Go in there and disturb what may be their nests. I found nests in luggage once, in my basement, as well as signs of mice eating bird seed. I now store that stuff in plastic containers.

Please do not lay poison! So many animals die from eating poisoned rodents. They are easy prey and then the animals which naturally predate on the mice can get poisoned too. Traps or a cat are the way to go. You can buy humane capture traps and if you get good ones you can capture the mouse and drive them out of town.

Victor Mouse Traps. Get the real ones not the knock offs. You want Made in USA snap traps. They are strong, sensitive, precise and quick. You geta quick kill in 0.012 seconds. Spend $20 and set out 16 traps. But before you do follow this procedure. It will make the traps even more effective.
Open the traps and let them air out for a day.
Put them in a bucket of bird seed for another day.
Use sunflower seeds as bait in half of them. Use peanut butter in the other half.
Place them along the wall but mix it up a bit. Put some facing the wall and some parallel.
Make a map so you remember where you put the traps and keep track of which ones are the most productive.
Put more traps where you are successful.

Don’t wimp out. I had them destroy a car by chewing on the wires and the interior.
They are relentless foes and reproduce quickly. Go at them hard, fast, and ruthlessly.
Good luck.

In addition to getting rid of the mice, you need to try to find where they’re getting in, and block it. They can fit through a hole the size of a dime. I second the snap traps; also, don’t be afraid to re-use them, the mouse smell will attract other mice. If you don’t care what your shed and its contents smell like, there are various mouse repellent products out there. (They say they smell ‘refreshing’.) Here is a page with a few good ideas.

By the way, if you’ve ever watched a cat kill a mouse, you know it is definitely not “humane”. The cat plays with it, tortures it, lets it run and re-grabs it with needle sharp teeth and claws until the cat is bored. Then the cat kills the mouse and leaves the body someplace as a display.
My neighbor’s cat brings live chipmunks and mice inside and drops them in the bathtub. It watches them scurry from side to side in the tub until they are exhausted. Then it bats them again to make them run some more. When the animal finally does not move after being kicked with sharp claws the cat finally kills it and leaves the body on the kitchen floor.
My neighbor raves “Oh isn’t Muffy a good kitty? She left me a present!” Gag me.
Give me a Victor Mouse Trap any day. It is quick.

Another vote for the Mouse Trap. It may look “cruel” but it’s quick unlike cats.

Here’s a good tip.

I had some guests over this past weekend, and one was telling me that she smelled an awful smell in her car. She took the car to the mechanic and he found that a mother mouse had babies in the fan of her car, and when she turned the car on, they all got killed by the fan and the mechanic charged her $600 to take the fan apart or do whatever to remove the mice pieces. @LuckyGuy makes a good point – the mice can destroy your car, so best to get rid of them asap.

And yet another vote for victor traps. They kill cleanly and quickly.

I assume that you can block all entry points.
The next step is to remove all food sources.
There are many live traps on the market: eaton multi-trap, mice cube, humane mouse house are a few you can google.
Quick kill traps can also be considered humane by some.

Are you rural? How about a couple of black snakes?