Why on earth are you even using 9”-12” wide cabinets? They are unusable from a functional perspective. Fewer larger cabinets cost less, and give you better storage space.

Attach them together properly before installing as a whole assembly, attaching into the studs with the proper cabinet installation screws. If the back is load bearing, you should be able to get a couple of screws into the studs. If there are internal hanging rails, you may be limited in attachment points, which should limit the weight that you place in them.

A lot depends on the physical construction of the cabinets, and you have provided no pictures of that.

You need to have studs to hold weight and IMO you will need to address that issue before installing any kind of actual storage. Maybe some wider open shelving where the shelves can be attached with more than one stud might be better also posting a picture of the space where these will go is a good idea.

Little update, this is a project house for me on a very tight budget...rental in the future perhaps up for sale, but the more love goes is (and by love I mean sweat!) the less happy i am about turning it over to "others". The original kitchen had only the one bank of cabinets which I want to save and repaint, so I searched Craigs for some similiar cabinets to flank the stove on a very short wall, you see it behind the stove, and to use on the frig wall. I found a tear out set for $20. I have several plans on how to use them, given the shortness of the wall there would be the super small ones next to the stove, but I like your comment so I will consider open shelving there, even that will not hit two studs so I may still have the same dilemma! I will be back for more layout questions/comments in the near future...yesterday the old vinyl floors came out...now how/what to do with the wood floors I revealed! Doing all the work myself so it's a slow process.