The dapple (or more correctly called Merle) gene is a pattern, so works on all coats and colours. The merle gene works by during development of the puppy, interrupting the melanocytes that are usually uniformly distributed through the skin to give the colour. When interrupted, we see the familiar dappling effect. The melanocytes are cells that produce melanin which is in hair, eyes and skin.
When written down, the gene is written as M, and the NON dapple gene is m. So your non dappled dog is mm. Dapple is dominant.

Dapple Mm (right) and non dappled mm (left) Dachshunds
Owned by Louise www.brengordachshunds.co.uk
If a dog inherits the merle gene M from one parent, and a non merle gene from the other, it will have the genetic code Mm. And will be dappled.
This merle gene can interrupt the pigment in the eyes, the same as the coat. This may give a wholly or partially blue or “wall eyed” appearance.

Full unilateral wall eye on a chocolate dapple longhair
"Fudge" Owned by Joy at www.middledach.webs.com/
Problems may occur if we allow a mating to take place where a puppy may inherit a merle gene from both parents, and so be MM. Merle is incompletely dominant. We only see the full effects of this gene in the homozygous state. But dapple CANNOT be carried. If a pup is dapple, so is at least one of the parents.

Chocolate "Double dapple", or homozygous merle.
photo provided by http://haagendasz.synthasite.com/
Double dapple dogs *may* have various deformities. Remember those melanocytes which this gene interrupts? Hair eyes and skin. So the double dapple has white patches, usually including a collie type white collar and white tail tip. This of course isn’t problematic until it happens to the hairs (or cilla) in the inner ear. Without pigmentation they are soft and incapable of processing sound waves for the brain to recognise as sound. So they can be deaf. The eyes may be affected too and in that case will be either small (micropthalmic) or just not there (anopthalmic). Other problems that the merle gene can cause in the eyes are irregularly shaped pupils and "notches" in the iris giving jagged or star shaped pupils. These irises cannot clamp down optimally in bright light and can lead to photophobic dogs.
Micropthalmia (reduced eye size). Modelled by Heidi, who is also partially deaf.
Picture courtesy of www.almosthomerescue.org

Anopthalmia (No eyes)
www.itchmo.com/itchmo-reader-dog-pictures-20-4868
(picture of pupil issues needed will credit and link)
So it becomes easy to see why we really need to know if our dogs are dapples! Not all dapples are bright, swirly marked. We bred one in the early 1990's with just a 5p sized patch on top of his head. Sadly I kept no pictures. But same as the brightest marked dapple, his genetic status was Mm and he was a dapple.
The merle gene works best on black based pigments - the eumelanins - black and its derivative, chocolate. Red or cream (the phaemelanins) dapples may never be clearly dappled unless they are shaded with black hair. Red and dapple ought not to be mated together, purely because of the risk if the dapple is not recognised. More than one breeder has come unstuck finding a red is actually a dapple. In all breeding, accurate pedigrees are a must.
Dapple pattern on a lightly shaded red dog. Almost invisible and only showing on the shading (eumelanin pigment) as opposed to the red (phaemelanin) which is almost completely unmarked.
Photo courtesy of http://www.freyasber.piczo.com

Heavily shaded red dapple, showing the action of the merle gene on the shading (eumelanin pigment) as opposed to the red (phaemelanin pigment)
Owned by Wendy Starkey, Hamoura Dachshunds.

This lovely lad is a cream dapple. Cream is a variation of the red pigment, so the merle gene only has a minimal effect on it, meaning that this dog is indistinguishable from a non merle.
These red or cream dogs who are dapple without it showing are not "cryptic merles" or "phantom merles" or any other name used to give them a rarity value. They are just merles/dapples without the pattern being visible.
Brindle is another pattern. It is not common here (UK) in Miniature Smooths apart from a few kennels who have this pattern through imported stock, but is common in Miniature Longhairs.
pic needed of brindle mini smooth

Brindle pattern on a cream mini long
Brindle is also dominant. Same as dapple, a dog must have a brindle parent to be a brindle. The only exception to this is if one or both parents are "e-reds". Please refer to the page on colour (unfinished) for an explanation.
Brindle works by disrupting the regular flow of black pigment and allowing it intermittently so the black pigment arranges into stripes. If however, the dog with this gene is a black and tan, chocolate and tan or dapple, you will only be able to see the stripes on the tan points.
pic needed of b&t brindle.
In the UK our breed standard does not accept white patches on dogs except for a minor blemish upon the breastbone. However, in the USA and Canada, piebald is a common pattern.

Piebald Mini Smooth
Picture courtesy of Nomistakin Dachshunds www.freewebs.com/nomistakin/
Piebald is NOT double dapple as is often mistakenly quoted. Piebald is the *recessive white spotting gene in its various forms. S is solid, si (irish spotting), sp (piebald) and sw (extreme white piebald) are varying forms of white spotting.
* Although mainly recessive, it is often possible to see the presence of the white spotting gene. Some solid coloured Dachshunds carrying piebald have white toes, feet or chests.

Ssw Dachshund. (Piebald carrier) The solid (S) gene is dominant, but sometimes incompletely so. Not all piebald carriers show white anywhere.
Picture courtesy of Nomistakin Dachshunds www.freewebs.com/nomistakin/
It is possible to have more than one pattern on a dog. A piebald may also be a dapple as shown below. This works as the dog is black and tan, then the two piebald genes add the white, and the merle gene interrupts the black areas to create dappling.

Or even a brindle
(pic of brindle piebald needed, will credit and link)