You catch it with moldy cheese from the castle dungeon, which is beneath the library.
Get the gold coin from the fountain and pay your admission to the Mordred Museum. Get the library slip from the stack of books and go to the castle library (forst floor, right). Press the shelf marked McM on the lower aisle, and the stairway to the dungeon will open. All you do there is get the cheese and throw the switch.
Go to the maid's room (first floor left) and use the cheese to attract the mouse.
Get to the basement by clicking on the Mcm shelf and the moldy cheese is down there then go back to the mouse and put the cheese on the floor and grab it.
To catch the mechanical mouse, in the Princess's Maid's room (castle first floor left).
Click on his card in your items bag.
It is a mechanical mouse. (see related question)
The mouse will get surprised, run around, then run back to you.
Feed it the mechanical mouse (from the Maid's room) and the owl will follow you.
After you get the cheese you put it in the princesses dressing room and capture the mouse to feed it to the owl.
At Mordred's secret lab, when the owl flies out, you feed the mouse to the mechanical owl. The owl becomes your friend and follows you.
the mouse is in the princesses room in the hole in her wall but you have to get the cheese to get him/her.
In the Maid's room (castle first floor left) you will find a mechanical mouse.
release it to the owl. then the owl will follow you
In the Maid's room (castle first floor left), feed the cheese to the robot mouse.
give him the robot mouse from the cell