1. When a seat emptied whoever spoke up first that they wanted it got it. This "system" was in place for small stakes games and to be honest works just fine.
2. A list is maintained at the table (usually on a pad of post it notes) and when a seat comes open the person highest on the list who wants said seat gets it. In theory that person is then cycled to the bottom of the seat change list, although in practice this just creates another grey area chock full of angles to be shot. Did he move? What order did they move in? Who's where on the list? He table changed and came back, shouldn't he be off the list? You get the idea.
So down here they employ a different "system", and I use the word very loosely. Whoever has been at the table the longest has priority for any new seats that become available. If you've been seated the longest, you can take every new seat that comes available until someone "challenges" you for the next seat. And I'm not even kidding, this old man (whose initials are BA, of all things) turned to me last week, after arguing with me over who would get the vacant 4 seat (I almost never argue for a seat with rare exception....if I am seated on the left of a man who does not chop in a game where chopping is coming up frequently, I have to move as quickly as possible) and eventually being convinced I'd been at the table longer than him (it wasn't even close....I'd been there for 4 hours, him 25 minutes) said "OK well I challenge you for the next one!" My face probably went expressionless as I tried to ponder what on Earth he was talking about before I eventually said "OK....sure" or some other such eloquent piece of prose. My new buddy (who doesn't yet have a stage name) was at the table and had to explain to me what was going on. I laughed and said to BA "You can have the next one sir, no worries". He promptly took my old seat (the funny part of this was that all I was trying to do was jump around him from the 3 to the 5 because he was the aforementioned non-chopper....later in the session when he posted a sb on the button he asked everyone if they wanted to chop and tried to back his way out of it claiming he only doesn't chop when there are 2 blinds but will chop 3 ways. This is completely backwards of what 95% of people do. It is common for people to chop 2 ways but not 3) and play resumed as usual, with him now 2 seats to my right. A little bruhaha ensued when the dealer tried to deal me in the very next hand (I wasn't entitled to a hand, I had to sit out one deal because I skipped one seat away from the button) claiming that the two of us had just changed seats and that if BA took a hand, I got a hand. This is actually the rule, any two players can change seats and if the one getting the worst of it (moving closer to the blinds) takes a hand the one getting the best of it gets a free hand. But technically we didn't switch seats, I changed then he slid over so I wasn't entitled to a hand and I just told the dealer to skip me and get the hand over with in less time than it would take for the floorman to come make a ruling. Anyway....
A few minutes later another seat came available and my buddy wanted to take it. BA was all over it, saying that he'd challenged for the next seat (he's actually using the word challenged) and since nobody had challenged him he was on top of "the list". But my buddy had been here longer than him....what was the protocol here? BA seemed to think he was entitled to the seat, which may have just been because he's 40 years older than my buddy, but my buddy really wasn't having it and just moved into the seat. BA got very upset, which is a pretty common occurrence for him since he plays for like 2 days straight and specifically yesterday bought chips at least 10 times ($200 or $100 at a time), and to be honest I don't even remember how the situation was resolved other than my buddy saying "You can have the next one, I promise".