Condolence and Condolences

When used in the singular, condolence generally refers to sympathetic sorrow, and particularly sorrow with regard to the loss of life. It is used when speaking indirectly of that shared sorrow:

After the 1985 Bradford stadium fire, during the Prime Minister's visit of condolence to the city, . . .
— William Leith, The Independent (London), 25 Nov. 1990

and often in a modifying position:

Still, although I felt for him, I barely knew the guy. Would it be inappropriate to pay a condolence call? Did they want their privacy?
— Alan Gelb, The New York Times Magazine, 7 Jan. 1990

Last week, people around the world showed their support for Russia. They held candlelight vigils, sent condolence letters and signed memory books.
— Jeremy Caplan, TIME for Kids, 17 Sept. 2004

The plural condolences, on the other hand, is often used specifically for an expression of sympathy, and most often appears in the construction my condolences, which is used to communicate sympathy. While condolences is often used to share sorrow over a death:

The prime minister of the day attended Hardy's funeral. . . . King George V and the Prince of Wales telegraphed their condolences.
— Terry Eagleton, Harper’s, November 2007

it can also be used of anything perceived of as a misfortune:

The odds of getting Apert syndrome are about the same as getting killed by lightning: 1 in 100,000. Indeed, for his family, Nate's birth was a lightning bolt: it came from the blue and was a shock to the system. Instead of getting congratulatory notes, the couple got condolences.
Bella English, Boston Globe, 1 Oct. 2000

Within mere hours of Ted Lambros's rejection for tenure at Harvard, communications began to pour in from every important university center of the United States. Some were simply to express condolences.
— Erich Segal, The Class, 1985

Condolences is sometimes used humorously, whereas condolence is not:

Tell someone you're from Buffalo, and you get the Look: If pity and condescension got it on, this is the baby they'd make. Something in the eyes that offers condolences for everything from playing childhood games in a frozen tundropolis to four straight Super Bowl losses.
— Nick Bakay, ESPN, 28 June 1999

Condolences is the more common form of the word, and should be used when expressing your sympathy at someone’s loss.