There are several great posts in the forums discussing various ways. I'll list a few that I know.
A dream journal is a good thing to start keeping, if you don't already. Sometimes our spirit animals make themselves known to us in our dreams. Focusing on the intent to find our animal(s) before sleeping may stimulate a dream that will help you in discerning this.
Meditation is also a very good way to try to figure this out. Spend sometime simply focusing on the matter, focusing on spirit animals and whichever animals you may think are yours.
Spirit animals tend to be the ones you feel attraction to, comfortable with, etc. Perhaps look up animals who fit this cryteria and read about their traits, symbolism, etc.