How do I hang/attach a microwave above my range with no cabinet to at?

EasIest way is a shelf. Either a floating shelf so no hardware shows or brackets with shelf on top. Hit the studs and check the weight capacity of brackets/shelving material to ensure it is safe and secure. Check the owners manual to be sure you leave enough ventilation around the oven. You could also fancy it up with underneath hooks for utensils, or make the shelf a shallow box work a drawer inserted for extra storage.

Just be sure you do a test run first to make sure there's room for the door to open and that's it's not too high to comfortably take out hot food dishes. You could put a lip to keep it more secure (I'm in earthquake country).

Hi Kristin, Over the range microwaves have a bracket that attaches to the wall. This bracket holds most of the weight. However, you are going to have to plug it in to a wall recepticle. If you are venting to the outside, you will want to hide the attachment to the vent as well. I would build at least a small shelf above the microwave so you can put decorative items to hide the socket. Before you purchase the microwave, be sure to get a copy of the installation instructions. A cabinet above MAY be required by the manufacturer for safe installation.

https://www.google.com/search?q=microwave+shelf+above+stove&tbm=isch&tbo=u&source=univ&sa=X&ved=0ahUKEwjtq_3ku6zcAhUOHqwKHSTyC0cQsAQIlQI&biw=1093&bih=498