My kids are dogs...we let them watch anything but they prefer anything dog centric. Or animal centric in general.
We were both raised to pretty much watch what we wanted, partner has an older brother and an older sister who liked to let her watch very scary movies at a very young age to "see her reaction" (they were quiet a bit older then her), now she loves horror movies.
For me I was allowed to watch anything as long as it wasn't too violent (no Terminator films for instance, my mom really can't handle violence at all), or something my mother hated. I still hear about how I made her sit through I Heart Huckabees. I remember being a kid and watching Beetle Juice, my grandpa didn't want me to watch it (he thought it'd be too scary), but I freakin loved it! Still do! And she took me to see American Pie when it first came out...I was...11 maybe 12...she didn't exactly know what it was about.
I'd probably any overly gory films (Saw franchise for example, I won't even watch them) are out right off the bat, but otherwise we'll see as we go along. I know when I was a kid my mother loved to use movies to "spark a discussion". For instance, The Little Mermaid. She gives up her voice to get legs and explore the world. What does this say about how our culture views women out on their own? Or when I got into Buffy, we'd discuss how one vampires not real, and then she made me read Dracula and research vampire popularity is the US. When I watched West Side Story with my grandma I ended up reading Romeo and Juliet...I was 9. Then I read Macbeth, the Taming of the Shrew, and A Midsummer Night's Dream. I'll probably end up doing something similar with my own kids.
One of my friend's kids, I do remember was terrified of...Toy Story. Specifically, Buzz Lightyear, when he was about 3. But he accidentally saw some of Texas Chainsaw Massacre (the remake), when his mom thought he was in bed, he just sat down and started watching. Cause Buzz...scary, Leather Face...nice friendly dude. Apparently.