In the olden days, audible ringing on cellphones was fine, because cellphones never rang. Most people didn't have one and the people who did didn't know how to turn them on. But now, in the terrible present, there's only one excuse for ever hearing your phone ring.
The only time you should have an audible ring is if you are expecting a call so important that if you don't get it, you or someone else, will literally die or have a baby without the father present. Basically, the only people who should know what their ringtone sounds like are paramedics. Otherwise, I do not want to hear your phone.
Hearing other people's phones ring sucks for two reasons. First, it's annoying and distracting. Second, it's disappointing: You think it's your phone, and when it isn't, you feel let down because no one called you. But that isn't even the main reason not to have your ringer on, since the whole point of cellphones is to make people feel like you're more popular than they are. No, the real reason not to have the sound turned up is this: When the time comes to choose people for the space pod to Earth2, and you have been a ring-a-ding everyone knows when you're getting a call type of person, you will be left behind to die a sad lonely death on Earth1 with only your stupid chiming ringtone to keep you company.
The ringtone, musical or otherwise, is dead. Mourn it, bury it and listen to Naughty By Nature on Spotify like an adult.
The sound-off rule goes for work, public spaces and home, unless you are at home alone, in which case it's okay to have a little ring. If a phone rings in a house and no one's around to hear it, no one gets mad at you.
It also applies to alerts. You may think it's fine to get a little jingle alerting you when joetaco44 checks out your OkCupid profile, but it is not. The literal thought everyone around you will have if your phone goes "tralala" every time you get a text or email is "I wish I could disinvite her to happy hour."
So, we've established when not to have an audible ring (whenever you're not a paramedic), but what if there's a call that you really need to get, like when Tina Fey wants you to do a writing test for 30 Rock or the pizza guy is at the door. In cases like that, it's time to bust out the vibrate.
Some people think that they won't hear their phone if it's on vibrate. But this isn't true, because vibrate is the loudest "sound" on Earth. When a phone vibrates on the East Coast, New Yorkers think it's an earthquake and Californians make fun of them. Vibrate is a desperate reverberation left over from the the Big Bang that becomes so deeply ingrained in your psyche that you start feeling the tremble of your phone even when it isn't happening. If you need to get a call, put your phone on vibrate, and then, and this is important, put it in your pocket. Never leave it out to vibe away on your desk, on the floor of your bedroom, or -- Oh, my God -- on the table of a restaurant. When your phone vibrates on a hard surface, it's as bad as if it were ringing with sound.