In general on piano do you play the melody of a song if you intend to sing? Is it the case that you play a much more basic version of a song if you intend to also sing? If I am looking at sheet music is this denoted in any fashion?
I believe this depends on the performer/composer. There are songs that the melody is just being sung and there are songs that the melody is being sung and played at the piano (or some other instrument).
Also, sometimes the melody is being played on the piano before the singer starts singing it.
If the composer wants to do something specific, this will most likely be denoted on the sheet music; otherwise, it's up to you to do what you like.
There is a lot of sheet music available which has 3 lines of manuscript. The lower two are the standard treble and bass clef that most people know from piano music, and the top line is often treble clef, and has the melody line, usually with the lyrics. Looking at the middle line, you'll find that the top line is often duplicated, but there are also other right hand notes to play.When a singer is there, the top part of the middle line is optional, it doesn't need to play the melody, because it's being sung. However, it can be played if the vocals and piano player so wish.
If you're working from real or fake books, all you'll get is the melody line and chord changes. You have the choice to play chords with the left hand, and melody with right, but if a singer is singing, it makes more sense to play chords and extensions, or fill in bits, with the right hand.
It has to be noted that much piano sheet music is not designed for accompanists, but for solo players, so has a tendency to include the melody in the top line, rather than being a literal translation of the original piano part [if any]
I would tend to treat this type of sheet music as a guide rather than an absolute, & work from memory as to what the original or favourite version of the song actually did.
I agree with everything everyone has said above. Just to add to what they've already said, sometimes its just better to play by ear. For example, if your wanting to sing and play an Adele song, like "No One Like You", you can replicate the same version if you just play what's on the recording. It is easier said than done, but you have a lot more flexibility if you develop that skill. Its better than being dependent solely on sheet music, because if you follow too adhesively to it, your at the mercy at some other musician's interpretation of the music. (Not to say that sheet music isn't great! Or a great place to start.)
A melodic playalong is mostly working well for simple music and simple interpretations. If you have a singing style with melodic and rhythmic freedoms, an accompanist will do the performance no favor by trying to follow the line along, even when self-accompanying. You also get the problem on a piano that it is a percussive instrument and thus has problems in delivering an even texture when long and short syllables are alternating.
In that case basically working on the chords, often arpeggiating them over both hands in order not to get a wham-wham-wham accentuation on the beats/changes tends to turn out nicer.
Piano extracts of music for choir and orchestra are not really helpful here since they are intended as rehearsal aids and thus integrate the singers as well. You see similar effects in piano extracts for solo music: they tend to remain perfectly recognizable even if nobody sings along.