I've been around the forum for years and I have never seen anyone with that problem exactly. What do you mean by "falls off" ? Describe the way you have searched again,if you would. Try to help us understand how you've been going at it.

the left rear signal lights literal short out and burn the wire toward were they go into the rear signal both were aftermarket LED of different brands and fell off while riding. I used a multimeter and found both the positive and negative are grounded by the ohms setting on the meter. I worked from the rear light back and the short is coming the set of wires that run from under the tank or the switch itself but the front left signal works, the set of wires that connect under the seat to the actual turn signal are good. it is a 2003 LS

What kind of LED lights/wires are you trying to use?

If the LED's you bought work as both running lights and turn signals, and the ground is through the mounting stud - then you are wiring them wrong by connecting one wire to ground and one wire to power.

If the LED's you bought are single purpose and only have one bright light function - then most likely one wire is ground and one wire is power.

The stock turns signals lights do not use the mounting stud as a ground, and this may be causing you a problem with your new turn signals.

If the chassis wires are not burning up, then take your volt/ohm meter and test the wiring harness without any turn signals connected.  You should have 12V of power available on the green wire for the right turn signal, and on the solid black wire on the left rear. (You will need to have the key turned on and the turn signal switch turned to the side you are testing).  The ground wires are the black wires with white marks.

Link to Wiring Diagram.
http://suzukisavage.com/cgi-bin/YaBB.pl?num=1294779818