Hi, you have face frame cabinets. Most likely, the open boxes for the cabinets are installed screwing to the toe kick / base (if the base is not constructed as part of the box) and a couple of screws to the back, from inside, and then you can see the odd screw to the cabinets to the side. Then the face frame is usually glued on the front, clamped, sometimes held with pocket screws from the back of the face frame, where you can't see it, or the occassional nail from the front, but the main frame support is actually wood glue... assuming 1980's cabinets, not 1940's.
The box cabinets sit on a base on the floor, (gravity) so don't need much anchoring except for tiltiing to the front and side to side levelling. They could also have glue and the occassional screw to the back wall if you are not seeing much. Stick built vanities will use a lot of glue on the wall, I know. One option is if the counter is off, at the back the boxes could have a support / spacer bar along the back top edge that is screwed through from above, at an angle into the studs, then the counter is put on to hide them. That longer bar / spacer spans multiple cabinets and helps to keep them lined up when the walls are unevern and offers a lot of rigidity.
It seems like the divider between two cabinet sections is fairly thin partition, and that your base cabinet is actually a double wide, with a thin divider and face frame front designed to make it look like two cabinets (or to hold the hinges for standard size doors). There would not be screws in this area.
