Can you have a singular "dreg"?

Solution 1:

However rare, dreg is indeed the singular form of dregs. The word enjoyed employment by Shakespeare in Troilus and Cressida (Act 3, Sc. 2):

What too curious dreg espies my sweet lady in the fountain of our love?

I managed to gather a few words ending in -s that are most often used as plural-only: scissors, shears, tongs, trousers, measles, series, gallows, species, thanks, clothes, bellows, outskirts, pampas, premises, pajamas, rabies.

Solution 2:

No, it is not linguistically incorrect to use the word dreg: http://en.wiktionary.org/wiki/dreg

There are many other pluralia tantum as mentioned in this (closed) question where I learned about 1000 English pluralia tantum and Mass nouns