The inside of a vanity is never an empty rectangle, so setting out starts from measured positions rather than a standard drawer stack. A U-shaped box wraps the waste on three sides; a notched box clears it at the back only and keeps more of its own structure. Either way the cut-out needs 20 to 25mm of clearance all round, because a box that only just misses the pipe will rub and eventually jam. The notch should be stepped rather than simply cut away — a box with a bite out of its base racks under load.
Changing the plumbing buys more storage than changing the joinery does. A conventional trap and a pair of stops sitting mid-height in the middle of the opening dictate every box above and below them. Swapping to a slim or offset trap and lifting the waste higher up the wall will often release a whole drawer, and moving a pipe is far less work than fabricating a complicated box around one that did not need to stay.
With the obstruction resolved, the fit-out follows use. A drawer presents its entire footprint when opened, where a door and a shelf present only the front row. The most valuable box in a bathroom is the shallow one at the top, around 100 to 120mm, because small items disappear to the bottom of anything deeper, and deep boxes need dividers or a bottle grid to stay usable at all. Converting the toe recess into a shallow drawer recovers 100mm of height a plinth simply encloses, and the fixed panel under an undermount basin can be hinged as a tilt-out tray.
Hair tools are the one case needing services inside the cabinet. An outlet in a drawer keeps them plugged in and off the counter, and in a bathroom a receptacle needs ground-fault protection wherever it sits. On heat, the useful measure is a metal or heat-resistant liner a hot iron can be set down on; a ventilation slot mostly helps a damp brush dry.