When do cats purr?

WAG Just like the ads say,
Contented Milk From Contented Cows,
Contented Purrs from Contented Cats!

Don’t be fooled, they are digging escape tunnels.

“Well, then, you see, a dog growls when it’s angry, and wags its tail when it’s pleased. Now I growl when I’m pleased, and wag my tail when I’m angry. Therefore I’m mad”. - Chesire cat

I believe it’s been established that the vibration from purring speeds the healing of injuries, especially those of bones. There are medical devices meant to have the same effect and that vibrate at about the same frequencies as purrs.