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.)
"...If a woman entitled to arms marries a man who does not have arms, she may continue to use her maiden arms, but on a shield not a lozenge and on a banner, but differenced by a small escutcheon of suitable tincture displayed either in the canton, centre chief point, or in such other part of the shield as marks of difference are customarily displayed."
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:
"Mr. Udal suggests that an armigerous woman who marries a non armigerous man may still display her own arms. But how? Her husband has no shield, so where are the wife's arms to go? She cannot bear them apart, on a man's shield, because she is not a man; she cannot use a lozenge, because she is neither maid nor widow. Of course if the non-armigerous husband dies, and the armigerous lady marries an armigerous male, then her arms can be impaled with her second husband's, or put in pretence, if she has no brothers.
"Not long ago a friend of mine consulted me as regards a friend of his, a married woman, who wanted an armorial book-plate of her own, apart from her husband's. This seems to me, as I told him, impossible. No married woman can hear her arms apart from her husband's shield, whether impaled or in pretence. The only (apparent) exception is when a peeress in her own right marries a commoner, and her shield, with coronet and supporters, is placed side by side with his. But even then her arms, ensigned with the coronet of her rank, are also placed in pretence on the husband's coat.
"The armigerous woman marrying the non-armigerous man does not forfeit her arms. They are there, ready for use when they can be used. As long as her husband is non-armigerous they cannot be used--cannot be in evidence. If the husband becomes armigerous, or she marries an armigerous man, then the impediment is removed." -- March 7, 1903
"...There can be no consistent reason why a lady of coat armour in these more enlightened days (when a married woman is considered a separate entity, and is allowed to hold property independentiy of her husband) should not bear her paternal arms on the feminine lozenge if she pleases, and that whether her husband be armigerous or not."
"It would seem that a lady married to a non armigerous husband should bear her arms on a lozenge, after the analogy of a peeress in her own right who is married." -- April 18, 1903
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:
"(3) A married man will continue to have the option of bearing his own arms alone. A ruling of the Kings of Arms made on 6 November 1997 allows a married woman to bear her own arms alone differenced by a small escutcheon. That will continue to be the case but the addition of the mark of difference will forthwith be optional."
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:
"In cause arms were first ordained that a mighty noble warrior might be known from others … if arms be given to a man and he hath issue a daughter, to what intent should she bear his arms with a difference? To none; because she shall never wear coat armour nor come to wage war in the field … Wherefore understand that she beareth arms only to be known of that noble blood and [for] the continuance of the right to those arms in them that shall come of her."
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.
I corresponded with the Chief Herald of Ireland's office on this question some years ago. My question related to arms inherited by an heraldic heiress, who herself had only heraldic heiresses (pre-Republic of Ireland), with that latter heiress marrying a non-armiger, and having male heirs. The Chief Herald's office said that such heirs possessed the abeyant right to quarter the coat of arms in question, should they petition for a grant of arms themselves.
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.)
In November 1997 the three kings of arms of HM College of Arms in London decided that a married woman whose husband was not armigerous could display her father's shield, differenced by an aptly placed and aptly tinctured inescutcheon. In 2014 they declared that the inescutcheon was now optional. The arrangement for married women is as above but with the difference provided by a lozenge.