can I put a microwave oven over my kitchen sink
Placing any electrical appliance near water should be carefully considered. You might want to check with your local code authority before proceeding.
As long as it is sturdy, I see no reason why you couldn't do that.
Ideally you shouldn't be able to touch water and electric at the same time, but I think this comes down to common sense, you know whether you will be tempted to use the microwave buttons with wet hands! My concern would be about bending down to do the washing up below the microwave and the impact that would have on my back.
You can if you really want to. I am assuming you are thinking of a moderate size counter top model. Are you going to hardwire it or will you use a plug and receptacle? Building codes require, "all counter area receptacles and all receptacles within 6 feet of a sink must be GFCI-protected".
I wouldn't. Water and electricity don't mix. Especially a 1500 watt + microwave. Can't imagine wet hands accidentally touching it or even water splashing on it. Recipe for disaster.
You might be able to fit a small one up there. In my first apartment I added a shelf over my kitchen sink between the two cabinets and used that for my small coffee pot to get it off the countertop because my kitchen was very small.
I would not do this as water and electricity can really hurt you.