Yup. With their maiden name only, of course: if Demi Moore is in private life Mrs. Ashton Kutcher, she could be called “Miss Moore” or “Mrs. Kutcher” but not “Miss Kutcher”. (She could use “Ms.” with either surname, though.)
When does it end?
Hopefully somewhere short of Tollemache-Tollemache de Orellana Plantagenet Tollemache-Tollemache.
Combining portions of last names could prove disastrous in some cases:
Furman and Black
Sherman and anything ending in “it”
Other examples are left as an exercise for the reader.
What is more confusing to me are those with two last names that don’t hyphenate. And it seems to be based on whim; Sacha Baron Cohen has a scientist cousin Simon Baron-Cohen and brothers Ash Baron Cohen and Dan Baron Cohen. Does Sarah Jessica Parker have one or two last names? No idea.
It’s also a pain for us substitute teachers in a heavily Hispanic area. (Re: Hispanic Surnames)
They give us a roll list where the student’s name is
LASTNAME FIRSTNAME MIDDLENAME, but only allow about 20 spaces for the entire string.
Many students’ official last name is a compound last name (not hyphenated), which ends up being so long that their first name is not printed.
Then the teacher has a seating chart with just first names, and no last names to help correlate.
So don’t blame me if I miss a student or two during roll call. 
My half-sister, who was from my mom’s first marriage, had the last name L.
Then adopted by my dad, last name then B.
Married, became J.
Divorced, went back to B.
Married, became A.
Divorced, went back to B.
Married, became H.
Not-yet divorced, but went back to B.
She could be KML-B-J-B-A-B-H-B. 
Among people I know, very few men have changed or hyphenated their last names. The only one I knew had the last name of Fink, and his wife had the last name of O’Dear, so they ended up as the O’Dear-Finks. But with most people it just seems like the wife changes her name and the kids keep the father’s last name, which kind of undoes the purpose in my opinion, because how “liberated” is that?.
It’s going to end when a Mr Cox marries a Ms Ucker…
A compound lastname or two lastnames? It’s not the same thing, a compound lastname gets inherited as a block; two lastnames get the lastnames from the other parent intertwinned. The lastname of Pilar López de Ayala’s father is López de Ayala (her second lastname is Alonso).
Yea, but many places still get it “wrong”.
UF- Never had a problem with hyphenating it.
LSU- Screwed up in some places not in others
UGA- Screwed up even more. What pisses me off here is that people can see my name and I’m listed in the directory under G (for Grenze, the first in the hyphenated combo). If you’re having trouble with my name, you can use the keyboards Ctrl C and Ctrl V and save typing the incorrect last name. 
Too bad I cannot say that to the ones who screw up.
What if the man is hyphenated - Dan Girly-Mann and the woman is also hyphenated - Stella Wine-Cooler?
Thirty responses in and no consensus. I guess this is an evolving issue.
If it were me, I’d either combine to form a novel last name or keep our names and work out a single last name for the kids. My preference (being a dude) would be for the children to take the father’s last name (either his existing or the new name him and the wife agreed upon).
Well, yeah, hyphenated lastnames were very uncommon until a few decades ago.
I agree up to the point where I met a woman who insisted that her preferred name was Princess. She was a real estate agent in Florida and sold multi-million dollar homes. I was a guest at one of these homes when she came by for dinner (and to schmooze her way into selling this house that was being considered for sale).
I told her unless she was lapping water and eating from the bowls over there on the floor, I wasn’t calling her Princess.
Try having to handle Arabic names. There aren’t any family names, and the tribal name is often omitted, or it’s so broad that everyone in the city has that name.
Arabs take the first name of the father and attach it to then end of their first name. The grandfather’s name comes third, followed by the great grandfather, and so on. At some point, they may attach a tribal name or something.
Example: Saddam Hussein Abd al-Majid al-Tikriti. Saddam’s dad’s first name was Hussein. Hussein was the son of Abd al Majid. They’re from Tikrit.
Now, Arabs are very proud of their family lineages. But newspapers have picked up the habit of reporting the third name of a person as the “last” name, even though the name could go on for ten or more generations. Fathers of famous people hate this because their name gets omitted.
Abdallah, son of Hamid, picks up the newspaper and finds that his son, Ali, has done something famous: “Ali Abd-Allah Hamid is so famous. Hamid did some great things. Hamid’s so amazing. Hamid will be at the Red Mosque at 1:00 on Tuesday…”
Western newspapers report this as “Ali Hamid…” Abdallah then proceeds to complain to Ali- “Why can’t they call you Abdallah? Why isn’t my name in there? Why do they drop it out? Can’t you do something?”
Add this to the fact that everyone in Tikrit named Basim who’s the son of a guy named Hamid will be named “Basim Hamid al Tikriti”, and you’ve got a terrible time figuring out who’s who. And that doesn’t even mention the kunya!
Suit yourself, but it’s a legitimate name. In fact, Prince (the recording artist) is actually named Prince Rogers Nelson.
I think a lot of the time the best thing to do when both parents have long names (hyphenated or not) and want to pass on the legacy of both sides to their children is to officially register one of the last names as a middle name on the kid’s birth certificate. I’m very glad my parents chose this option so I didn’t get saddled with a last name of 16 letters plus a hyphen.
Some of the aristocratic families in the germanic countries would hyphenate a lot. Prince Philip was born to the House of Schleswig-Holstein-Sonderburg-Glücksburg. (Although a House name is not quite the same thing as a surname.) I could have sworn I read about some German aristocrat with a nine-barrelled name. My google-fu is failing me at the moment, though.
On a simlar tack, at the school where I work there is a girl with both a double-barreled surname and first name: Sindy-Barbie Taylor-Smith*
Grim
*Fictitious name to protect the innocent
You obviously don’t live in Quebec! (See Hari Seldon’s post)
Was it Johann Gambolputty de von Ausfern -schplenden -schlitter -crasscrenbon -fried -digger -dangle -dungle -burstein -von -knacker -thrasher -apple -banger -horowitz -ticolensic -grander -knotty -spelltinkle -grandlich -grumblemeyer -spelterwasser -kürstlich -himbleeisen -bahnwagen -gutenabend -bitte -eine -nürnburger -bratwustle -gerspurten -mit -zweimache -luber -hundsfut -gumberaber -shönendanker -kalbsfleisch -mittler -raucher von Hautkopft of Ulm? 
From the Burns and Allen Show: Guest starring Frances Langford
Gracie) I’m Gracie Allen, of course that’s just my stage name, in real life I’m Mrs George Burns
Frances) Yes, I know, like in real life, I’m Mrs Jon Hall. I’m married to Jon Hall and you’re married to Geroge Burns. It’s the same thing.
Gracie) Not even close
All I have to say is that hyphenated names are bull$#!+; and it’s even worse when men do it. For God’s sake, either change your name or don’t – don’t do BOTH!
And if there’s no other reason for my opinion, here it is (scroll to the penultimate person pictured at this link):
http://www.neptune.k12.nj.us/11361042510950/site/default.asp