Absolutely, it trains them to "freeze" and wait, it can be very helpful in a emergency situation. Say the dog gets out the front door and is running around the street, get the dogs attention and telling them to "stay" forces them to freeze to that spot so you can grab them safely. A knife or something dangerous falls to the floor, put them in a down stay and you can safely remove the dangerous object without them getting in the way.
For me it helps distinguish from "wait" where I want to go first and then call them over. With "stay" I don't call them over; they have to wait there for me to come back. It's also useful when I want them sitting as the 'default' if they want something and I don't want to have to release them each time they do it on their own. Mine will sit in front of me if they want some tasty food I'm holding, for instance.
How old is your dog? I prefer laying an obedience foundation using food and prey drive/toy. I don't use stay but use a release command such as "free" or "okay" to let the dog know that he can stop the sitting, down or whatever behavior. With the release, you can again offer a lot of praise for being successful. Then you gradually increase the length of the behavior, such as the sit or down, while adding distance between you and your dog. Eventually the dog will be well trained and there is no need for a stay command. It is just another word that has no meaning to the dog and it is difficult to convey the meaning of stay and more efficient to use a release command.
With the fosters, if I have time, I do teach a "Stay" because most adopters will expect it and so the pup might as well know what it means.
With Pongu I initially taught "Stay" as a separate command for the same reason that I initially did not teach an automatic Sit at Heel when I stopped -- because we were training in freestyle at the time and you do things a little differently in that discipline to get a smoother-flowing routine.
Currently Dog Mob does not need a separate "Stay" command, they just hold position until I tell them to do something else. Sometimes I give it anyway for clarity though; it's basically a secondary cue that I sometimes find helpful as a reminder under stress. "Wait" is a separate thing to them; it basically means "do not pass this doorway until I say you can, but you don't have to hold a particular position in the meantime."
I'm working on getting Pongu to Stay during heeling when I lead off with my right foot and come with me when I lead off with my left foot (i.e., standard competition obedience footwork) but we're not there yet.
Several times a day. For example, before she gets out of the car I tell her to stay while I pick up her leash to attach it to her collar. If my dog starts walking too far ahead of me, I tell her to stay until I catch up. When I'm at our office, I'll have her stay at the top of our outdoor stairs while I walk to the car, open it, and load some things inside. Sometimes during training, I might be running my dog through a course and then unexpectedly may need to stop (due to a loose dog) and stay is used as a "freeze" as Shade described in the 1st response.
I use hand signal for stay. I do sit and then hold out my hand or down and hold out my hand. I also work on duration of sits and downs so that she will sit or down until released, but it's a habit for me to hold up my hand because when I see her wanting to move and I'm across the room I can remind her silently with the hand that she's supposed to stay put
I'm with a few others, no double commands. Sit means sit until you are released or told to do something else. If he leaves his spot I put him back and tell him again that same command. My release is not as easy as "free" I wanted it to sound different so I use "lets go" when I release him.
Yes, it would need it in life and in competition, agility, or schutzhund. It is always better to ask your dog to sit and remain in sitting position in any awkward situation with another dog, they calm down faster whilst sitting or lying down. You may need to walk into some shop, to some object or person leaving your dog waiting for you. Start training him patiencer by walking away from him short distances first, then longer, walk around him, ask him to wait longer. Ideally, your dog shouldn't change his position until a new command was given. Train him patience when throwing the ball, throw his ball, ask him to sit for couple of minutes, then send him after the ball. "Stay" is normally supported by gesture of your hand outstretched like in "Stop".
Without knowing it well your dog wouldn't learn "Freeze", when he must freeze in the middle of the motion, all in all, "Sit/ Stand/ Down, Fetch + Stay" works as intermediate command before learning some more difficult commands. You simply stop pronouncing it after a while when your dog starts to understand that he must remain in whatever position he is until you tell him something else. It demonstrates how well you are in control of your dog.
Since in AKC you're allowed a second "stay" command. Most people training for AKC ob/rally will have a "stay" command. In Schutzhund...second command is not allowed so they don't train it.
At this point in my dog's training (he's three) I usually don't have to tell him to stay if I just said "sit" or "down" but on the long down/sit exercises I'll say it just to make 100% sure he does it. What I've been taught about obedience trials...if you're allowed to do it, do it.
"Stay" is kind of a broad command in our house. I don't use it for sit or down, the dogs are expected to drop, no matter what they're doing, and aren't allowed to release themselves until I give the command. "Stay" is really for when I am walking out of the house and they have to stay inside, or I'm getting out of the car and they have to not jump out. I don't like the idea of yelling "stay" if the dog is running away. Then the dog is left standing, sitting, downing, whatever.... it just seems more confusing to me than dropping into a "down." And I've had to use the down command in mid run and jump. Once to protect an elderly neighbor from my excited nine month old GSD galloping towards her and once to stop my corgi from jumping after a cat lol.