"on (his OR her OR my) last legs" - 72,400
"on (his OR her OR my) last leg" - 21,600
Amer. Heritage Dict. of Idioms:
Extremely tired, close to collapsing, as in 'We've been cleaning house all day and I'm on my last legs.' This hyperbolic expression originally meant "close to dying," and in John Ray's 1678 proverb collection it was transferred to being bankrupt. Soon afterward it was applied to the end of one's resources, physical or otherwise. It is sometimes applied to things, as in 'That furnace is on its last legs.'http://www.answers.com/topic/on-one-s-last-legs
leg - ...
on one's last legs - near the end of life, usefulness, or existence: 'the foundry business was on its last legs.'
I'm not aware of ever having heard the expression 'last leg' except in the context of a race with several stages or laps, or a relay race. As far as I know, if you say the runners were on their last leg coming up to the finish line, it doesn't mean they were about to keel over from exhaustion (though that may also coincidentally be true), but that they were in the final segment (= leg) of their race. I suspect many of the apparent web hits may actually reflect this kind of usage, and others might just be typos, but perhaps someone else could check that.
If a significant number of people really do use it in the singular, it could still just be one of those mistakes based on a misunderstanding, like 'another thing coming' instead of 'another think coming'. I wouldn't put it in LEO, though, unless support can be found for it in some printed dictionaries.