Although others work well, this is one I've written myself and am willing to share. Before starting anything, calm yourself. I usually do this by lighting a candle and meditating for a half hour or so. Sometimes more depending on How I feel. After meditating, I snuff out the candle with my fingers, (Never Blow) and continue on with the spell I'm carrying out. In this case, finding your Spirit Animal. After meditating go outside during the day. If you can get to a forest or even a quiet park this is great, if not, your backyard should work fine. Find a nice cozy place to sit on the ground where you can be relaxed and take in nature, I prefer taking off shoes and socks and digging your feet into grass. Take a few deep breaths, close your eyes and begin to chant, "Spirit Animal who knows me well, come and listen to my spell. Lead me to the distant path, lead me onward, home at last." try not to be rushed when chanting, a loud voice is not needed either. As you chant, become one with the nature. After a while, you should be visited by an animal. Something in the flesh. Talk softly to the animal, beckon it to come to you. If it truly is your spirit animal, and the time is right, it will come to you without fear, and once you have touched it, you will immediately understand the connection as if you were the animal.
If it meets in the flesh wouldn't your spirit animal be limited to those creatures which live wild in your area?
Also, a word of caution with the whole digging your feet into the grass thing. In domesticated places (like backyards) and certain wildernesses this is fine, but in a lot of others you'll find yourself digging your feet into pointy rocks, burrs, and slivers of plant material to jab under your toe nails. Always check the terrain before trying that.
My spirit animal is the Falcon, when I discovered this I was in an area where the falcon that visited me is not natural. Afterwards it followed me home and nested in one of my trees, something also very strange. I see her various times during the day and she is very protective of our home. She leaves "gifts" like a cat would and I am occasionally asked if the Falcon is a pet. On the matter of thorns and such, yeah you really need to watch for those, my first time was in a meadow, so I had no problem.