Quick answer
Red dragons go with craks, green with bams, white (soap) with dots. Dragons are never interchangeable, and a soap only becomes a zero when the card says so, at which point it stops being a dot.
Dragons look decorative and are not, which is where the confusion starts. Each one belongs to a suit, and once you know which goes with which, a whole set of card lines stops looking arbitrary. The rule is short: red with craks, green with bams, white with dots.
The twelve tiles
There are three kinds of dragon and four of each, so twelve in a set. Red, green, and white, the last of which almost nobody calls white at a table. It is the soap, named for a face that is blank or lightly framed and looks like a bar of soap.
On the card a dragon is written as a D, and the colour it is printed in tells you which dragon is meant. That is the same convention the card uses for suits generally, which is why colour on the card is information rather than decoration.
Each dragon belongs to a suit
This is the rule that makes those card lines readable, and it is easy to play for months without noticing it. Red dragons go with craks. Green dragons go with bams. White dragons, the soaps, go with dots.
So when a card line pairs numbers in one suit with dragons, the dragons are not free choice: they are the dragons of that suit. A hand of bams wants green dragons. A hand of craks wants red. Get this wrong and you build a hand that looks finished and is not.
It also explains the phrase you will hear at the table, opposite dragons. When a line uses numbers in one suit and calls for the dragons of the other suits, the dragons you need are the ones belonging to the suits you are not using.
Dragons are not interchangeable
Worth stating plainly, because it is a common misconception: a soap may not be played as a green or a red dragon, and it may not stand in as a dot tile inside a run of numbers. Each dragon is its own tile, and the suit it belongs to does not make it a member of that suit.
The practical version: three dragons of mixed colours are three unrelated tiles, not most of a pung. Beginners sometimes hold all three hoping they add up to something. They do not.
When a soap becomes a zero
The soap has a second job. On card lines that need a zero, the year hands most obviously, the white dragon stands in for it, because a mahjong set has no zero tile and years since 2000 keep needing one.
Two conditions matter. First, a soap is only a zero when the card shows it that way, not whenever you would find one convenient. Second, when it is acting as a zero it is neutral: it stops being the dot dragon and belongs to no suit at all. That is why a hand can pair a soap-as-zero with numbers from any suit without breaking the suit rule.
One limit follows from this and catches people out: a soap acting as zero cannot be part of a run of consecutive numbers, because runs are built from the number tiles one through nine and a zero is not among them.
Dragons and jokers
Dragons follow the ordinary joker rule rather than a special one. A joker can stand in inside a pung, kong or quint of dragons, and can never stand in for a single dragon or inside a pair of them.
That matters more than it sounds, because a great many card lines ask for a pair of dragons rather than a pung. A pair of soaps is a pair like any other: you draw both or you do not get there, and no joker will rescue it.
Playing them
Early, a single dragon is a weak holding. It serves only the lines that want its exact colour, and until you have chosen a direction it is a tile with one job and no evidence that job exists. Most of the time it is a comfortable pass.
A pair of matching dragons is a different animal, because pairs are the bottleneck of most hands and a natural pair of dragons opens the honour lines and the suit lines that pair with it. Hold that.
Late, dragons become dangerous discards for the usual reason: an opponent showing an exposure of numbers in one suit is visibly interested in that suit's dragon, and it is the easiest tile at the table to predict a need for. If somebody has bams exposed, think twice before letting a green go.
Put it into practice
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Knowing which dragon belongs to which suit is the easy half. The harder half is spotting, in your own thirteen tiles, which lines that unlocks. Mei, our AI coach, reads your hand and says which direction it goes for and what to keep, dragons included.