What markings you have access to is dependent on two things, their Variant Lines, and what traits they inherited from their parents. Only certain variant lines show certain markings, so if your geno has a mark that isn't a marking from the Primary or Secondary line, it will be carried and will not be visible on the design.
Markings are broken up into 4 groups. "Purebred" markings, "Combo" markings, "General" markings, and lastly, "Free" markings.
Purebred markings are markings that can only be obtained and exhibited on a geno that has the same Primary and Secondary variant line.
So, for example, you breed two dragons with a Primary and/or Secondary in common, then there's a chance the offspring will have an identical Primary and Secondary, and also present with a Purebred marking.
Combo markings are markings that are only present in genos with special combinations of Primary and Secondary Variant lines.
For example, Combo Mark X only shows on genos with A/B Primary and Secondary Variant Lines. So you cross one parent that has A/A Primary and Secondary, and one that has B/B Primary and Secondary. The offspring has A/B Primary and Secondary, and thus has a chance to develop Combo Marking X.
General Markings are markings that are further subdivided into different Variant Lines, and can only present as long as either the Primary or Secondary Variant line of the geno matches the line the marking belongs to. If one of the lines does not match up, then the marking is carried and does not present on the design.
For example, General marking X is a part of Line A, so as long as the geno is A/- or -/A then it will present the marking.
Free Markings are markings that anyone can add to their designs! These aren't included in the design's geno/pheno, and thus aren't inheritable, or passable, but you can still use them! You can use as many of the free marks as you want, or none at all!
A special thing about Purebred and Combo markings is that even if neither parent has it the geno still has a chance to develop it, unlike General markings.However, dominant Purebred and Combo markings can only present when at least one parent has and passes down the marking.
Carried Markings
Sometimes, your geno may inherit a general marking from one of it's parents, but it doesn't have the variant line the general marking is a part of. If so, then the mark is called "carried" and does not show on the visual design. For example, the parent has General marking X, which is a part of the Variant Line A. The geno inherited Marking X from it's parent, however the geno has a Primary and Secondary Variant lines of B/C. So, in this case, it would be classified as a carried, and would not be shown on the visual design.
Physical Traits
In DoK, each family line has a unique appearance, with special traits that go with that. Occasionally, when breeding two different family lines together, it may result in the Secondary Family Lines passing down a physical trait that wouldn't normally be visible. Certain mutations also have a higher chance of occurring when you mix and match family lines. But even besides the occasional mutation, there's also several traits everyone is free to add to their geno's import! This can help give an extra personal touch to your designs.