If it's a pet, I would just bury it. Would you throw a family member in the trash when they die? It's just a dead body, but it just feels appropriate to do something more respectful.
If it's like mice or something, or a bird that smashed into your window, then yeah I just put it in the trash.
One time I found a dead crow in my yard while mowing the lawn, so I picked it up and put it in the same bag I put the clippings in. That bag somehow got forgotten while all the other bags got brought to the dump. This was at my parent's house where my dad insisted on bagging grass rather than just use a mulcher, so we had like 40 bags. But that one bag got left behind.
A week after when I was mowing the lawn again I noticed the bag. I did the unthinkable, and opened it. "There's room in here" I thought. The smell of a decomposing bird, mixed with the smell of wet grass, made for a very weird smell.