Should you be able to see someones vein protruding more and more from the side of their neck whilst they're singing?

Like this you mean?

There is no easy answer to this ...

The SCMs (sternocleidomastoid-muscles) definitely have their function in singing, especially in neck anchoring. They don't have a direct effect on the voice, but an indirect one when it comes to support. High energy singing or high notes will be much easier to sing if the neck anchoring (which doesn't only involve the SCMs) is stable. So a certain visibility of the SCMs and maybe the neck veins is no reason for concern if the voice production feels easy and not strained.

If the veins only swell however because of straining and too much subglottal pressure that the cords cannot (yet) cope with, then no, it is not a good sign.

If the vocal production feels easy though and the sound is good, a slight visibility of the SCMs and the neck veins is usually not a problem.

I'm with Mezzo - there's no scientific evidence to suggest that seeing someone's veins is a bad thing. Each body behaves differently and some have more visible veins than others. Neck anchoring is an essential part of healthy singing (sadly not frequently taught) and so this may cause some neck bulging.

If the singer is red, with veins popping out all over the place, however, there's a fair chance that they're using too much breath - a common problem - the breath is probably the least important part of singing (it's just the fuel - not the process - sure you can sing without it, but it doesn't merit the ridiculous amount of importance it gets in traditional pedagogy - often to the detriment of the larynx and pharynx, which actually make and shape the sound...) - so if a singer has been pushed to think of breath as being everything, they may indeed over breathe, push too much air at the folds (particularly on high notes, which in fact require less breath, not more) or - in the case of rock singers - push a lot of air when singing in a twanged configuration - potentially very harmful.

Have you got a video of the singer in question? I reckon we could tell you very quickly if we saw him or her !

No video.
It was a girl 12 yrs old, really lovely voice, sang with her mouth almost closed so goodness knows how lovely her voice would have been if she opened it! She sang sideways on to me and I was a aware of seeing her neck vein.
It didn't sound strained, but she didn't look relaxed either.

It's hard to say to be honest, then.

If it sounded free and pleasant and she didn't appear strained in any way, it was probably ok - but without seeing her it's difficult.

One thing is certain, you shouldn't be relaxed when you sing - certain muscles need to be working very hard (particularly the neck and back in certain voice qualities).