There always seems to be more stuff that belongs under the bathroom sink than there's ever room to put it in. Vanities don't have shelves under them because the drain and water supply lines are in the way. My solution works around them, and leaves the shut-offs and sink trap accessible.

Step 1: Tools and Materials

Required tools:

Pencil
Tape measure
Needle-nose pliers

Optional tools:

Drill with bits (only necessary if the inside walls of the vanity are melamine or other hard or slippery material)
Combination (honestly, a plain level would work; in the absence of either you could just wing it)
I didn't end up needing the screwdriver.

Materials:

Screw-threaded hooks, 8
Light chain, 4 times as long as the width of your vanity (cable or wire might work, but I couldn't get rope taut enough)
Not shown: storage bins the correct depth to fit under the vanity

Step 2: Step 1: Measure and Mark

Basically, we're running pairs of chains across the inside of the vanity in order to establish planes on which storage bins can rest. Virtual shelves, if you will.

Tailor the measurements to your specific vanity. Start by holding a chain up across the width of the vanity. Move it up and down until you find a place where you could put another chain behind it to establish a plane. Measure that height, and draw horizontal lines on the right and left inside walls of the cabinet. This is where the combination square comes in handy, but as long as the lines are basically horizontal, you'll be fine.

To create a virtual shelf the full depth of the cabinet, drill pairs of holes at 1/3 and 2/3 of the depth of the vanity. Be careful not to drill all the way through the side. Drive the hooks deep into the holes on the left side, but only deep enough to catch properly on the right side. The needle-nose pliers help to crank the hooks in place. Again, don't screw them in so much that they puncture out through the sides of the vanity.

To create a toilet paper shelf, establish a plane just below the bottom of the basin. Drill one pair of holes an inch away from the back of the vanity. Drill the second pair of holes 3 inches from the back. The standard 4" American loo roll will perch neatly there.

Step 3: Step 2: String the Chain

Using the needle-nose pliers, cut the chain to about an inch less than the interior width of the vanity. Hook the left end of the first chain over the left rear hook, then string it across to the right rear hook. Screw the right rear hook in until the chain is taut. Repeat with the other chains, moving back to front, and right to left. Finally, make sure that all of the hooks are lying horizontally, so that they don't catch on whatever you rest on the chains.

The toilet paper gets tucked up into the awkwardly-shaped space beside the basin, up inside the top of the vanity. Be sure not to obstruct the works of the sink stopper.

Step 4: Step 3: Pack It In, Get It Out

Congratulations! You now have three layers of storage where there was only one before. Storage bins allow you to pull items out as needed, and let you empty the whole thing out quickly (for instance, if you need to close the shut-off valves NOW). If you need to do serious work under the sink, unscrewing the hooks is the work of a moment.