Any foldable stroller is allowed. As long as it's a normal sized jet, they can take it. I've seen massive chariots sitting at the door of United's aircraft!

The only real headache are those strollers with wheels that had to be removed but I think that was more an issue in the 80's when I worked. I don't think too many of those are around anymore.

Bag your stroller if you want but make sure the tag can been seen. Using a bungee cord is a good idea to make sure it stays shut on route. Pay attention to the handles. If you have foam or anything that could get ripped, you may want to either remove the covering or wrap the handles. You can even throw some plastic and tape on them at home and walk through the airport like that (avoiding doing anything extra at the airport).

Mine were once ripped (either UA or LH).