The history of the Mulligan:

1.) Magic was originally published without a mulligan rule.

2.) The first official mulligan rule was introduced by the DCI in 1994: (known as the original mulligan)
A player with an initial hand consisting either of all lands or no lands at all could reveal their hand, shuffle it back into their deck, and then draw a new, full hand of seven cards. This process was allowed only one time per player. (“all land/no land”-condition !)

3.) A new rule was tested out 1997 in Boston, and finally instated during the Paris Pro tour, hence it is known as the Paris mulligan:
Without any prerequisite, a player could take a mulligan if he wasn’t satisfied with his starting hand. The player didn’t neew to reveal their hand. The player had to shuffle his hand back into their library, and draw a new hand with one card less than prior to the mulligan. Multiple mulligans were allowed, up to starting with 0 hand cards.

3.a) For Commandergames, the „Partial Paris Mulligan“ or simply Partial mulligan or Partial Paris became popular, though it was never an official mulligan rule:
Instead of putting their entire hand back into their library, a player may choose to put a number of cards from their hand aside face down, then draw that number of cards minus one, after which they shuffle the cards they set aside into their library. The big difference is that not the entire hand is shuffled back, but players may chose which cards are to keep and which are set aside, AND that shuffling is executed after all the mulligans, ensuring, that the set aside cards are not to be drawn again for the starting hand.

This is the Mulligan we use as Teamcommander Mulligan. It was the most popular mulligan since now. The single difference you might notice is that in Teamcommander, the first mulligan is for free, but this is not a big exception enforced by us, but the official rules for Teamplay (2HG) by WotC, granting a free Mulligan in teamplays.

4.) The Vancouver mulligan rule was first tested at Pro Tour Vancouver:
The process is also very similar to the Paris mulligan, with the addition that any player who keeps an opening hand with fewer cards than their starting hand size may scry 1. Like the previous Paris mulligan rules, this mulligan is intended to reduce the frequency of „non-games“, where the winner is effectively determined by the contents of the players‘ opening hands.

5.) The London mulligan, which is the official mulligan at the moment, was introduced in 2019.
To take a mulligan, a player shuffles the cards in their hand back into their library, draws a new hand of cards equal to their starting hand size, then puts a number of those cards equal to the number of times that player has taken a mulligan on the bottom of their library in any order.

In other words:

• Draw seven cards every time you mulligan.
• When you are satisfied, you put a card back on the bottom of your library for each time you have mulliganed.
• No scrying.

6.) Also far from official, there is the Gabriel Special Surprise, or GSS Mulligan:
Each player initially draws 10 cards, then selects 3 cards that they do not want and shuffles these 3 cards back into their library.

People often stick to old times and mix up mulligans now, still believing that the actual mulligan includes scrying by example. Other players think you get a free mulligan whenever you have 0, 1, 6 or 7 lands in your starting hand (which is 50% of all conditions regarding 2,3,4,5 lands doesn’t grant you a mulligan), which is of course also not true – the idea behind this was just that 0,1, 6 or 7 lands are mostly not considered a good starting hand in a competetive play, and you should reconsider taking a mulligan.