A rented bathroom has two rules: nothing goes into the tile, and it all has to come off again. That rules out most of what the hardware store sells, but it leaves a surprising amount, because modern adhesive holds far more than the suction cups it replaced. Here is the room in four zones, in the order that gives the most relief for the least effort.
Before anything: prepare the wall
Every adhesive failure we hear about happens in the first ten minutes. Clean the spot with isopropyl alcohol, not bathroom spray (cleaning products leave a film that repels water, and your adhesive). Let it dry completely. Press the mount on hard for a full minute. Then leave it alone for 24 hours before it carries anything. Mount on the flat face of a tile, never across a grout line, and skip textured or matte tile, which gives air a path under the pad. If you want the reasoning, suction cups, adhesive, or drilling covers it.
Zone one: the shower wall
Bottles on the tub rim are the first thing to fix, because they are the thing you knock over every morning. A corner is the easiest place to reclaim: Nook is a coated-wire basket on four clear adhesive pads with an open grid so water drains instead of pooling. It publishes its size but not a load rating, so keep it to everyday bottles. If you want two shelves and a bit more capacity, CoveRise is a pair of aluminum corner trays with movable hooks, rated by the maker for up to 22 lb each when properly mounted.
Zone two: the basin edge
The vanity collects toothbrushes, the hair dryer and whatever has no home. A slim wall tray takes the small things: Ledge comes in three widths with an optional towel bar, and RippleClear does the same job in clear plastic when the room is small enough that a black shelf would feel heavy. The hair dryer gets its own hook: HaloDock cradles the barrel and takes the cord on two lower hooks. Toothbrushes stand apart in Lustre, which needs no mounting at all.
Zone three: the door and the wall beside it
Towels are where renters most miss hooks. TuckLoop holds a corner of any hand towel in a silicone slit, so nothing needs a sewn loop, and ClingArc is four matte-black hooks you can group or spread between rooms. Both come off with a hairdryer and a length of floss when the lease ends.
Zone four: the floor and the drain
The drain is the smallest use of adhesive in the room and possibly the most useful: MeshGuard is a fine mesh you trim over the drain that catches hair and then peels off with it. A non-slip mat such as Loom or Drift finishes the step-out, and neither mounts to anything.
What still has to be drilled
Be clear about this, because it is a safety question rather than a preference. Anything you might grab to steady yourself — a grab bar, a rail an older relative leans on — has to be screwed into studs or proper backing. Adhesive is not rated for the shock of a falling adult, and we would rather lose the sale than sell you an adhesive grab bar. Ask your landlord; most will allow two properly fixed screws for a safety rail.
The short list
Everything above, with who it suits and who it doesn't, is gathered on one page: no-drill bathroom storage for renters. The wider bathroom collection has the pieces that do take screws.