So I'm playing on a very old Harmony baritone ukulele. It has not been treated terribly well, and it is probably from the 50's. I'm having two problems.
First, the lowest string (I play plectrum tuning, so its a wire-wound string tuned to C3) will be in tune when open. Gradually it gets more and more sharp as I go up the fretboard. By the time I get to the 12th fret, it's so out-of-tune that it's unplayable. +30 cents at least. I suspect this may be some degree of warping of the neck.
The more pressing problem is that the second fret alone is very sharp. The first, third, fourth, and fifth are pretty close but fret 2 is way sharp. Is this due to the fret being worn down? I do use it a lot as I play, considerably more than the others in that neighborhood.