I wanted to think about what I thought was a drain catch, but I guess a drain catch is something in the pipes/drain under the sink. I mean that piece that keeps hair and such from getting in the drain. I went to get my haircut, and interviewed my stylist about her opinion on drain protectors (my term). Here is the drain catch she uses, which is a bigger version what we use at my house:
I call it a wire mesh drain protector, to be more specific. She said she doesn't like this style because you have to pull it out and clean/scrub it out to get it clean.
At home, she uses one that is stainless and inverted, so it is like a little stainless hill on top of the drain, with a stainless rim, and larger punched holes. She prefers it because you can pull the hair off easily. It does let viscous liquids such as toothpaste and bodily fluids to get into the drain more easily, which can build up and clog the drain.
I find it interesting that a shop that deals with hair all the time would choose the wire mesh type. I also find it interesting that the sinks don't come with some sort of special drain protector installed into the hair washing sink.