Nothing says "newly trained PADI diver" as loudly (or as lamely) as the snorkel firmly affixed to the mask band at the beginning of a dive, every dive. At least in this part of the world, here in the Indo-Pacific. But that's just one PADI-ism that many (not all) divers outgrow with experience and/or training with other agencies.
In certain conditions, wearing a snorkel while diving can actually be dangerous. I've seen people holding on to a rock or mooring line in a strong current have their masks dislodged, flooded, and even completely blown off due to the drag of their snorkel attached to the mask strap. Not fun. These weren't beginning divers either. I've also seen people lose their masks on a back roll entry, for the same reason. Fortunately, losing the mask and snorkel on a backroll entry is usually only embarrassing, as opposed to flooding or losing a mask in a powerful current, which can lead to a truly dangerous and even life-threatening situation. (And yes, they weren't holding their mask and regulaor properly on the back roll entry.) In the category of students, I've seen newly-minted divers practicing removing and replacing their regulator start to put the snorkel into their mouth underwater instead of their primary or an alternate second stage, which is a surprisingly easy mistake to make, especially for someone unused to diving, their equipment, or who could be slightly narked. Not that they'd have much of a hope finding the spare 2nd (octopus, I think PADI divers call it, inexplicably) if they really needed it, since on beginning PADI divers the spare reg usually seems to be stuck in a BC pocket, dragging along the reef, or else stuffed into a scumball clipped out of sight and out of mind on some corner of their BC.
Of course, snorkels can be very important pieces of safety gear in some situations. You will definitely be glad you have yours (and a compass!) if you ever have to undertake a really long surface swim. Just don't wear it on your mask strap when you aren't breathing through it (i.e., on the surface). You'll also want to have a snorkel with you if there is an opportunity to jump out of the boat and into the water with dolphins or other cetaceans on the way back from your dive, so bring it with you when you leave the mother ship even if you plan to leave it in the tender while you're diving.
Where to stow the snorkel, if you aren't going to strap it to your head? (This is for after your PADI dive course - not during):
(1) Strap it to the calf of your leg instead of that 14 inch stainless steel Rambo combat assault knife that you planned to buy after finishing your dive course. You won't need that.
(2) Try a folding snorkel that will fit in a BC pocket. (But I've never found one that didn't leak, or that I really liked);
(3) Fix the snorkel to the side of your backplate with a couple loops of shock cord. (This is what I do on open ocean dives where a snorkel might be needed, but this only works with a back-plate-and-wing rig. With a brandname jacket-style BC, you'll have to figure out something else.)
One more thing - a snorkel is one piece of gear where you really do want to select brilliant, day-glow colors. Bright yellow and red are good. Some SOLAS reflective tape wrapped around the top of the snorkel in two or three stripes near the top won't go amiss either.
It's been a long, long time, but I'm pretty sure I was not required to wear a snorkel when I did my basic open water training with CMAS, many years ago. I can also attest that IANTD doesn't require snorkels worn on any course dives. Some IANTD instructors I know might even refuse to let you take their course if you tried it. But I have never heard of anyone doing their basic open water training with IANTD, although I think a course exists, at least on paper.
PADI's dominant position in dive training is basically a reflection of Gresham's Law in action in the dive training marketplace - "Bad money drives out good". But it doesn't matter. Go ahead and take the PADI basic open water course. Yes, you will be required to wear a snorkel during the class sessions. Just do it. No need to embarrass your instructor by arguing that this practice can actually be unsafe - she/she probably understand that very well already. Once you finish the PADI course, then you can forget about wearing the snorkel on your mask, and start learning to dive.
- Frogfish2 (OK, I did take a nitrox course from PADI. But only because I needed the damn card.)