Of course, you know your little one best, but usually 15 months is young enough to not have fears of "scary" things yet. The fears of scary stuff didn't set in with either of my two until around the age of 2 years. My daughter was 19 months on her first trip and my son was 14 months on his first trip. We had no problems with fears from either one at those ages. At 3-1/2 though, my daughter was afraid of her own shadow, lol! We would never have even attempted haunted mansion on that trip!
You should be ok on any of the rides without restrictions, however, I would still test her out on the milder rides first (like peter pan, small world if it is open, pooh, etc) and work your way up. You just never know how a child will react. Many of the rides are dark, which didn't bother my kids at that age, but you never know.