When applying the brakes on my vespa px the lights go out and the rear brake light doesn't work

The high mount brake light gets its own signal from the brake light switch (on the brake pedal). Or at least the wire leaving the brake light switch will splice, one wire to the high mount brake light, and the other for the rear brake lights is routed through the turn signal switch, and coordinated there with the turn signal wires to the rear brake lights. The brake wires and turn signal wires share the same wire, and same filament in the bulbs. It is kind of complicated, but brakes applied with no turn signal, brake light wire goes from brake light switch through the turn switch and out to the bulbs. If brakes applied with turning signal, non blinking brake light wire goes from turn switch to rear light, while the turning signal wire will go through the flasher relay, and then to the rear light to cause blinking.

I hope that makes sense. It appears the problem may be in the turn signal switch for both the turn signals and the rear brake lights.

bad earth the wire that goes to the car body could be lose

it's not a bug - it's a feature....some people would say...
- actually it happened on mine too after some time when it was new on both brake levers, that means it cannot be a short or something.
but, after some time I got used to it and even got to like it like that.

if you have experience as a mechanic you can do it by yourself, but otherwise it is better to have it done by a pro, as the brakes are a main safety item....
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The chances are that your brake light switch is jammed on. The switch is connected to a spring, which in turn, hooks onto your brake pedal. There is a picture of one of these units available here: http://www.ebay.co.uk/itm/MOTORCYCLE-YAMAHA-TYPE-REAR-STOP-SWITCH-BRAKE-LIGHT-/310231989451 if you are unsure of what you are looking for. Two likely causes

1) The spring is over tensioned. There is an adjustable thread on the switch which allows you to move it closer to, or further away from the pdal, relaxing or tensing the spring. Release the tension on the spring by adjusting the switch unit so it is closer to the pedal (But test that when the brake is applied, the light DOES come on).

2) The unit is jammed as a result of age and or corrosion or dirt. Clean the unit, unclip the spring from the brake pedal and pull the centre pin of the switch in and out repeatedly to ease the movement. This may get it working again. If exercising the unit manually for 5 minutes doesn't fix the problem, you may need to replace it.

It sounds like the rear brake pressure switch is bad. The wires for the front brake light switch are wired in parallel to the this switch. Replace the pressure switch in the metal brake line coming from the rear brake master cylinder going to the rear brake caliper.

Good Luck
Steve

check the connection of the bulb ensure itis fitted correctly also check fuses it maybe that the bulb for the lights has blown, it is common that the brake light is on a seperate fuse for safety.

Could be a deficient battery, malfunctioning solenoid, starter is burnt, starter switch, or brake light switch, as these are necessary. If your brake light is not working, that's it. I need more information.