Sundays in Paris are still considered a day of rest, for religious reasons - that is, the French faith and belief in time off work. But if you live with neighbors in close proximity (and especially if you have parquet floors, like most Parisians) you'd better not spend your day off mowing the lawn, catching up on vacuuming, or installing some new shelves - that is, unless you want your neighbors thumping on the ceiling to subtly and oh-so-passive-aggressively point out that they don't like the noise you're making. (Thanks to Mazarine for that anecdote!) Some Parisians have even been forbidden by their neighbors from running their clothes dryers on Sundays.
You can also scratch off that other UK and US Sunday tradition, of reading a fat newspaper filled with cultural supplements, coupons and cartoons. Most French newspapers are relatively thin, and as I type, they're actually on strike. And if you need to do any shopping, you're not having a lazy sleep in - shops that are open on Sunday usually open in the morning (say, 9-1) rather than the afternoon. Exceptions include areas market 'tourist' - this includes the shopping mall in the basement of the Louvre and the 4 Temps, a big mall at the end of Metro line 1. Food shopping after 1pm is usually limited to gas stations and the corner shops, which are relatively pricey and may not have any fresh bread.
My recommendations instead? Church, of course - in French or in English (St Joseph's RCC, St Michael's Anglican, American Church in Paris, etc). MCM and I usually like to eat - heading to the market in the morning, buying supplies, and then cooking in the afternoon. We also usually try to counter the eating by getting out for a nice walk or a bike ride. You can now rent Velibs near pretty much every Parisian park, including the Bois de Boulogne and Vincennes.
Or, and here's my top suggestion, try one of the markets that are open in the afternoon. I like the covered, open-air book market in the bottom of the 15th arrondissement, which is open every weekend and specialises in used and antiquarian books.
Bring cash. Most books are in French although there are some in English. Some of the vendors are very specialised - like antiquarian books on French colonialism - and some carry jumbled bits of everything (Andrew Morton's biography of Monica Lewinsky, anyone? Or the Hachette Guide des Vins 1987?) There are sections dealing with children's books, cookbooks, fine art books, science fiction, and (gulp) erotic cartoons. If you're just looking for a pocket version of any French literary classic, you'll find it here for a euro.
When you're done, there's a little restaurant across the street, Les Tontons, which specialises in tartares, or there's a Poilane bakery. You can also take a nice walk in the Parc Georges Brassens, a pretty modern park that has lots of play areas for kids.