Pick a wire, any wire - Intermittent electrical fault
 

Every so often, but not all the time, I get a buzzing noise from the drivers foot well when I press the brake pedal.

With the help of a friend stood behind the car I have confirmed that when I am getting the buzz I am also not getting the rear passenger side brake light. At other times the brake lights are just fine.

The left indicator works fine at all times unless I am pressing the brake pedal and getting the buzz in which case it flashes double time on the dashboard as if the bulb has blown.

Somewhere there is a snapped of chaffed wire. I am guessing that there will be a wire with a break in it still held together by the plastic wire surround so it will be invisible and being an intermittent fault and absolute git to find.

If it was just the brake light I could check the wires to and from that but the indicator adds some mystery to me. Earth wire from lamp to car?

Any suggestions on where I start and what order of checks I make?

I get a buzzing whenever the ignition is on, quite a loud buzzing from the engine compartment. It stops about 5 seconds after the ignition is turned off.
It's not intermittent it is always there and I have no idea what it is.

Mine is specifically when I press the brake pedal, just like an on/off switch

My (few) experiences with auto electricians have taught me to do minimal fault finding and just rewire the offending circuit. Providing a dead short is NOT involved, that will solve the problem even if it doesn't clear the cause.
Inter-related stop light and indicator faults usually involve bad earths.
It might be worth checking the battery drain for each of your circumstances to make sure that only the expected amps are being used when each circuit is activated- need that friend and a meter between the +ve battery terminal and its lead.

The earth for both brake and indicator go into the same connector inside the car.

I wiggled them and sprayed WD40. Problem went away.

Fixed it.