I have put this together as something of a play-aid for those playing in the current Craftsman's Reach campaign (Fenrus' Very Best, as they are currently known). For that reason, it will focus on the kinds of PCs that are actually in that game, but some of the general concepts can be applied across the board in the 10th Age.
The Big Difference: Intensifiers and the all-purpose Fuck
The intensifier and exclamatory FUCK does not exist in the 10th Age, because swearing is modeled after a classical/medieval mashup. Obscenity is a part of life, and there are no blushing violets who will turn away from engaging in the swearing language. Fuck is used as an obscenity, but only as a verb -- "The gods fuck me," for example.
What, then, are the swearing conventions, if you can't just pile up fucks on things?
A lot of them are explicitly sexual in nature, even more so than the now-denuded "fuck" and "fucking." Being fucked by other things in nature, including the gods, is a common complaint. Being pissed on is as well—"Fortune pisses on me," for example. Rather than exclaiming "fuck", someone in the 10th Age is much more likely to exclaim, "I am fucked!"
A swear that does carry over from the modern world is "Fuck me!" though it is often embellished a great deal ("Fuck my ass").
Hell and damnation is a very different set of terms in the 10th Age. For one thing, "hell" is not a unitary place, but rather a realm with seven layers. One does not tell people to go to hell, but one might exclaim "Seven hells!" as an exclamatory curse.
Common curses refer to the sacral and genital parts of various deities and even regular folk. Male gods, for example, are commonly referred to by their testicles and penis ("Heimir's hairy balls!", "By Haeron's cock!") and goddesses are referred to by their own generative organs ("Vaela's cunt!"). More proper or uptight folk might refer to other identifying marks of the gods ("By Heimir's barrel!" "By Haeron's hammer!", "Star and stone!", etc.)
"Piss", "pissing", and "shit" are all common swears in the 10th Age.
Since there is such a taboo on interacting with dead bodies in the cultures of men, the most obscene combination one could utter would be a sexual or bodily act involving the dead. This would be highly insulting to dwarves, who revere their dead as ancestral spirits.
Certainly raunchy priesthoods might very well indicate that, when things go bad, their gods are fucking them. Heimir, Raya, etc., are gods wherein their priesthoods would have no issue with exclaiming, "Ah, Heimir fucks me!"
Dwarven swearing is inventive and frequently sexual, graphic, and violent. For male dwarves, this often focuses on their pride and joy: their beard. "Shit on my beard," is a particularly vile phrase. Dwarven humor can focus around performing sexual acts with the family members of the other person engaged in the joke. Dwarven threats sometimes involve acts of violence, beards, and family members ("I'll dip my beard in your boy's blood", "I'll tear your father's beard out by the roots", etc.).
Female dwarves focus more on the violent aspect, often referring to the hands and generative organs of male dwarves. ("I'll flatten your prick", "Draw that blade and I'll break your hands", etc.).
Wind elven swearing is generally inventive and referential. They often refer to past elvish histories with gruesome content, making allusions to characters in the elvish story-cycles. This can lead to a complex tapestry of content.
Wood elves swear in a much more straight-forward way, complaining about the various qualities of trees in relation to body parts.
Gnomes and Halflings
These races generally take on the swearing conventions of whoever they are near.