Swap bulbs after cleaning contact surfaces.
Check for good ground circuit.
Check for good power circuit.
Only need to check from point where wires separate to go to either lamp.

 

The fog lights use one fuse and relay for both lights.

Since the relay and fuse are in the engine bay fuse box, fairly close to the lights themselves, your problem is most likely a cut or pulled wire.

What year is it?

 

Go to the dead power circuit, connect a test light to the battery positive cable, follow the circuit testing the wires along that circuit, especially at any connection points. Once you find power, you have found the problem.

My suspect would be the lamp itself based on the running the car without the lens intact to seal the socket from corrosion.

 

It's a 2003 Corolla 1.6 VVTi.

Thanks for the advices, I'll throw an eye ASAP and keep you up to date

 

Since 03 uses an H3 fog light bulb, I would also have a real good look at the socket.

 

Well, it's working... but I don't get why :

As you can see on the picture, there are 2 sockets. I was certain that the bulb was initially plugged to the black one, so I didn't bother to check with the brown one (I was a little in a rush at this time).
Today I thought that It would be very stupid if indeed I made this mistake, and yes, the good socket is the brown one ! :grin:

However, why two sockets ?
There is 0 volts coming from the black one. Could it be a repair that the precedent owner did?

Hardtopte72 : it's actually a H10 bulb, not sure why, since the car is supposed to have H11 bulbs... Could the black socket be a H11 ?

In any case, thanks for the help ;-)

 

I just realized you aren't in North America. For US and Canadian vehicles, 2003-2004 used H3 foglights and 2005-2008 used 9006 fog lights. 2009+ use H11 foglights.

I don't have a good answer for your connector situation.