I'm aware this isn't very time-specific, and not sure if the right SE, but what happens when a woman who belongs to a noble family and has a coat of arms, marries someone without a coat of arms etc.?
Modern practice is this, according to An Heraldic Alphabet (p. 231, 1996 edition) by former Clarenceux King of Arms, J.P. Brooke-Little:
(edited to add -- this was a new addition to the 1996 edition, mentioned as a change in the heraldic laws.)
A lozenge is a diamond shape, traditionally used for women's arms in some cases instead of the shield. An escutcheon is the shield shape. So "differenced by a small escutcheon of suitable tincture" means that she would display the arms with a small shield shape of a suitable color added, in the upper corner, the upper center, or elsewhere.
In 1903 the topic was discussed in letters to the publication Notes and Queries. This is long, but it's a fascinating look at some slightly less modern opinions:
A bit more about the modern Law of Arms in England and Wales (Scotland has its own separate heraldic authority) is on the College of Arms website.
Another thing that might be relevant -- the College of Arms recently posted a ruling on the arms of individuals in same-sex marriages. It's relevant to this question because of this section:
So the tiny escutcheon is now optional, I guess.
This is only peripheral to the OP's question since it doesn't address the non-armigerous husband issue, but there is an article, "Heraldry and the Medieval Gentlewoman", in the March 2003 issue of History Today. It does discuss some interesting aspects of heraldry in the female line and might be of interest. Unfortunately, the article is behind a paywall, but you may be able to read it online through your local library.
The article's author, Maurice Keen, quotes a fifteenth-century lawyer saying something very similar to the words of the March, 1903 letter writer above:
The answer depends on the country and the era. You've limited the question to England, and implicitly to the pre-modern period. The best resource is probably DeBrett's, but it isn't terse, it isn't easy to use, and it isn't free. I'm going to summarize with broad generalizations.
Titled nobles are armigerous (have the right to arms). A given Noble may have multiple titles (he may be the Earl of X and the Marquis of Y, and the W of Z). If you don't have a title, you are just a Lady or a Gentleman.
A father needs to ensure that his children have a living or an income. Since nobles cannot have a profession or engage in trade, that living is one of four forms (1) land & title, (2) military (3) clergy, (4) marriage. Fathers have a limited number of titles and lands; the normal option is to give the bulk of the titles & land to the eldest son. The second son joins the military, and the third & successive sons join the clergy. (Later in history the father would need to settle some lands on the second son to ensure that his military career would prosper as an officer, but that's outside the scope of OP's question).
Daughters cannot join the clergy or the military, so they are limited to either marriage or titles. Every title granted to a daughter diminishes the patrimony (and power, and wealth) of the eldest son. The eldest son also has a customary responsibility to advance and protect the welfare of his siblings, so it is a bad idea to diminish their welfare.
So to wrap up, although a father may choose to grant a title & lands to a daughter, that is usually done as part of a dowry to make a better marriage. HOWEVER, if the daughter is the only child, then the titles may devolve to her (and then convey to her husband) at various points the crown may need to approve the conveyance. In general Scottish titles are easier for a woman to inherit. Here is a list of titles that convey in the female line
OP asks what happens if an armigerous woman marries a man without arms. The answer is probably going to depend on politics. If the man is common, then her titles are probably forfeit - because a baseborn man has no chance of protecting the titles or serving the king (he is almost certain to have no training in arms, and nobody will follow him in war). If the man is noble, then her titles probably convey to him and to their children, but that will only happen if he can defend the title (again this part depends on which era in history. If we're talking Game of Thrones/War of the Roses era, then EVERY noble must be prepared at all times to defend his lands and titles. Kings permit their nobles to carry out limited warfare among themselves because it ensures that he will be supported by the strongest nobles in the event of a national war. (I suppose one could consider the king to be pursuing an optimal portfolio strategy of military strength).
Given the breadth of the question, this is a collection of unsupported generalizations. If I could narrow it down, and if I had lots of time, I could find specific references & examples. If you want to read further, try the wars of the roses (which George Martin admits is one of his sources for Game of Thrones). Alternatively look at the history of Empress Matilda, which is brilliantly, if fictionally covered in the Brother Cadfael novels.
It is also worth consulting She-Wolves (As an American, I haven't seen this yet. Waiting on BBC America) which is based on a book (I have a copy in the "to read" pile, but based on an interview with the author, I think it will go a distance to revising our understanding of women in English history.)
I'm sorry, but Mark's answer strikes me as almost entirely wrong. He says that nobles cannot have professions etc - I don't believe that to have ever been the case in England, though it was in some continental countries. He also says that the father "gives" his lands and titles to his eldest son, younger sons join the army or the clergy - well not necessarily!
Generally, nobles had little or no freedom as to whom inherited the patrimony and certainly none over who inherited the title/s - that was set by the terms on which the title was bestowed, and almost always "in tail male" - that is, the eldest son got the title and most of the land, with some provision for daughters' dowries. Whislst a father could choose to grant unentailed lands to a daughter, he most certainly could not grant her any of his titles. Even now in England there are very few titles above baron that can be inherited by a woman - a fact many male peers feel unfair and outmoded.
On the position of an armigerous woman marrying a non-armigerous man, I have to confess I don't know, and my book on heraldry does not touch on it. However, it was common in the mediaeval period for men to claim titles "by right of his wife" if she had no brothers to inherit; John of Gaunt became Duke of Lancaster by marrying Blanche of Lancaster.
I think there is some confusion here between bearing a title - Baron or above - and having a coat of arms - not every armigerous family is or was noble. Also, the position of an heraldic heiress is slightly different, referring to all the daughters of an armigerous man, not necessarily noble, who have no brothers. In this case the wife's arms are displayed on her husbands as an escutcheon of pretence - a small shield in the centre of the husband's arms - as he is "pretending" to be the head of his wife's family. Any children - male and female - would then quarter their arms with both parents' as opposed to just their father's.