The basic harmony is usually a third above (or a 6th below, which is the same note an octave lower. Example: in C - sing a C - the harmony will be an E, either the one directly above, or directly below. So here, we can call it a major third above, for simplicity. However - in key C major, the note that harmonises with D will be an F. this is because they're both from the C key - diatonic. Technically, it makes that interval a minor 3rd, but so what? It sounds good, and doesn't suddenly make the song go into a minor key!
When you need a harmony for an F note (still in key C), it'll be an A - straight from the C scale, as that A is a third above the original F.
'When someone sings an F diez(#)' it will depend on which key you're in at that time. If it's the same song, in C, then I'm guessing the underlying chord is a D major/seventh, and thus an A natural will fit best.
To summarise - use notes usually from the key/scale at the time. Use notes from the harmony used at the time, which may be different, but usually still diatonic.
If you're just singing along, really, you shouldn't worry about whether it's a major or a minor third you're actually singing - your ear ought to tell you. Maj/min 3rd is more of academic interest - and a heck of a lot of folk who harmonise spontaneously won't even be aware what notes they are singing - except that they'll be the right ones, judging by the best judges we use - our ears.