This may have been discussed somewhere, but I can't seem to find an explicit statement on the confidentiality of one's AHV/AVS number anywhere... not even on the AHV website.

Would anyone know if this is a piece of information one should not be giving out lightly, as is the case with the social security numbers of other countries?

May I ask why? Is someone asking for your AHV/AVS number or do you want to gather these from a phishing scam and sell them on to the Nigerians ?

What does the letter say that your AVS card came with (I don't have it with me at work)?

It's being asked for a gym membership.... together with my address, health insurance details... just about everything but the colour of my underwear.

I can understand this is a number that is given out to an employer... perhaps even to a bank when asking for a loan. But I'm not understanding why a gym might need it, or if it's normal to give it out... I'm afraid I don't have the letter with me (also at work).

Probably so you get a discount on health insurance.
The only 'bad' thing they could do with the no is to set you up on their payroll & pay you some money......