I agree, it's probably from the old scales which balanced each other. (ie. like the Libra sign) These are actually called "a pair of scales". Much like we say "a pair of pants" or simply "pants", even though you could argue that it's a single item. ;)

It's possible to use the singular "scale." "I get on the scale to weigh myself." "I used the kitchen scale to measure the flour." I use the plural "scales" to describe that old-fashioned type of device that has a "T" with strings holding two "plates." Also, commercial trucks that are used for shipping sometimes have to get on the scales to make sure they are within the legal amount of weight. Those scales are underground, so I don't know what they really look like.

We usually say it singular. "The scale". But in some figurative expressions, we say "tipped the scales in his favor". It is due to the fact the 'scale' in older times was actually two parts, like Peachy mentioned

You can use it singular like BRSwartz says but I have to be honest it sounds a little strange and I dont think to be honest if we said this to ourself in our head we could honestly say thats how we would say it...I just tried and found myself putting the s on the end, I get on the scales to weigh myself, I used the kitchen scales to measure the flour...no saying in the singular just isnt coming out, it needs to be plural. Im not sure why it is this way maybe because scales tend to have two or more types of measurements on them. Or perhaps from old fashioned scales that have a tray on each side for putting things in to weigh. The one time you can defiantly use it as a singular is to refer to a musical scale, ie please play the c major scale for me!