Here's a trick I learned from my Peruvian ex...but it's not furbaby related, hope that's okay.
If I drop oil on my shirt, pants, couch (yes did it a few weeks ago), I put TONS of baby powder on the oil stain right away and let the powder soak up the oil. Then after an hour or so, or until you see the powder all caked up, I scrape off the powder and put the clothes in the laudry. And...Viola! No more oil stain. Works everytime...even on my brand new couch!