Currently there are two ways to accumulate a total of 60 free Cheese.

Typing /facebook in-game will bring up the game's facebook page and will award you 20 Cheese. Give it a like while you're at it!

Clicking the Menu icon below the bottom left of your mice's avatar next to the chat will bring up the game's menu. Clicking change your password and following the steps will award you with 40 Cheese and a special item, the Cheese Earring that can be equipped through the shop in the 'Ears' section.

Typing /atelier801 will award you with the Cake Hat, celebrating Transformice's 3rd Anniversary. The item can be equipped through the shop in the 'Head' section.

Doing a special combination of emotions on Map 0, or in your Tribe House, will award you with a badge, celebrating Transformice's 4th Anniversary.
You can join a room and wait for Map 0, join your tribe's Tribe House, or join /room #utility0[Your Mouse's Name] and play Map 0 by typing !map 0
Press these emotion buttons to the right of your avatar: Confetti, Dance, Confetti, Clap, Kiss

Logging in to the game via Steam will award you with a badge.

These are no longer possible to obtain.

Logging in to your account using Atelier801's mobile app "Run for Cheese" will award you with a badge.