There are six basic patterns of inheritance of single gene traits found in humans. These patterns depend primarily on the location of the gene (which chromosome it is found on) and the alleles' effects with one or two copies. I will describe each of these six patterns. I am refraining from describing the results of all possible Punnett Squares for the sake of brevity, but if you want to work out the specifics for each case that is the tool to use.
Autosomal dominant: An autosome refers to all chromosomes in the nucleus that are not sex chromosomes (X and Y). A dominant allele is one which has a phenotypic or observable effect when an individual has at least one copy of it. This means that an individual with an autosomal dominant phenotype could either have one or two copies of the allele while an individual who does not show the phenotype must possess no copies of it. For an individual to inherit this phenotype at least one of the parents must also have this phenotype.
Autosomal recessive: An autosomal recessive allele is one which only influences the phenotype of the individual when both copies of the gene are this allele. An individual with an autosomal recessive phenotype always has two copies of this allele. These traits can skip generations, meaning two individuals who do not have the phenotype but both have one copy will have a child with the phenotype 1/4 of the time.
X-linked Dominant: An X-linked gene is one that is found on the X chromosome. Because females have 2 X chromosomes and males have only 1 this results in different inheritance patterns and phenotypes between the sexes. A dominant allele will create a phenotype in both males and females. A male with this trait will always pass it on to his daughters, but will never pass it on to his sons. A female with this trait will pass it on to half or all of their offspring depending on whether they themselves have one or two copies. This trait will be more prevalent in females than in males because females will be more likely to have at least one X chromosome with this allele.
X-linked Recessive: A recessive allele on the X chromosome will affect more males than females. This is because males only have one X chromosome so a single copy of the recessive allele will be sufficient to create a phenotype. Females with two X chromosomes will require two copies of the allele to present the phenotype. Males will never pass this trait to their sons, and they will only pass one copy to their daughters which is not sufficient for the phenotype to present. Females with this trait and therefore two copies of the allele will always pass it on to their sons, while females without the trait but with one copy will pass it on to their sons half the time.
Y-linked: As humans typically only have up to one copy of the Y chromosome all alleles on the Y chromosome can be thought of as dominant. Females cannot present Y-linked phenotypes as they do not have Y chromosomes. A male with this allele will always pass on this phenotype to his male offspring.
Mitochondrial: Humans actually have a small additional genome outside of the nucleus in their mitochondria. This mitochondrial genome is only inherited from the mother via the egg and not from the father. As a result, mitochondrial alleles can also be thought of as entirely dominant. Males do not pass on their mitochondrial DNA to their offspring but females always do. Male and female offspring will have exactly the same mitochondrial DNA as their mother.
These are the basics. In reality, most human traits are the result of many genes acting in concert and are referred to as complex traits. Each individual allele is inherited according to the above principles, but since many alleles affect the traits it is difficult to determine the effect of each individual one. Most traits are also heavily influenced by their environment. This leads to the concept of penetrance which is the idea that even for single allele traits the presence of a phenotype and its strength can be a stochastic distribution for reasons not always well understood.
EDIT: To address the specific examples you gave in your question and generally on the plausibility of inheritance patterns that don't neatly fit into the above categories it's important to recognize that genetics is incredibly complex and is full of weird phenomena and bizarre exceptions. With enough knowledge and thinking you can probably justify just about anything you can imagine, but that doesn't mean that whatever you come up with is actually likely to ever occur. I'll take a shot at providing an explanation for each of the examples you gave.
trait W has some probability $p$ to occur given that the mother has it with certain probability, but no information is available about the father and the general prevalence among males is $p_m$.
As far as I can tell there are no difficulties with this scenario. It could describe any of the automosomal, X-linked, or mitochondrial examples patterns above. If you'd like to calculate the actual probabilities the fundamental equation at the heart of human genetics is the Hardy-Weinberg Principle.
trait X should only be observed in females, but may be passed on to the next generation silently by males
If this trait is related to being female in some way this is trivial. Males can pass on predispositions for breast cancer for instance but are unlikely to present it themselves. However, if this is a general trait that would be expected to be observed in both males and females it is difficult to justify why it would not present in males. Some possibilities might involve X-inactivation or genomic imprinting.
trait Y is not systematic, but if one individual has it, then it must follow that their sibling have it too
The genes that one individual inherits will always be independent of those their siblings inherit. While generally, we say sibling share 50% of their DNA, that's only an average. In reality, siblings can share as little as 0% and as much as 100% of their DNA although either extreme is quite unlikely. The only genetic way to ensure that all offspring of a set of parents will have a trait is if the trait is recessive and both parents have it. Another explanation could, however, be environmental. If, for instance, a mother exposes all of her children to the same odd prenatal environment that could cause all of them to possess a similar trait.
trait Z is rare, but if both parents have it, their children are very likely to have it too.
This trait would fit nicely into an autosomal recessive pattern of inheritance.
