For $1000 to work with, there's plenty of options for decent bikes. Personally if I was buying a complete single-speed and has a grand to play with, I'd buy a Raleigh One-Way and use the remaining money on any upgrades I wanted to it. Or, with $1k, you could probably buy a decent frame and piece together a good singlespeed on your own if you're up to it.
For that kind of money, check out Trek's District line. The top model has a belt drive that is dead silent and very fun to ride.
Masi has a Speciale Commuter that's very versatile. It'll take fenders and racks, it's got brakes.
Bianchi San Jose has a flip flop hub so you can run singlespeed or fixed. Can take wider tires if you've got bad roads.
Check out IRO or the Specialized Langster or a Surly frame and build it up yourself. Cannondale, Giant, Felt, Fuji, etc. all make singlespeed bikes. Test ride a bunch, read all about them. Get the one that's the most fun for you to ride and ignore our recommendations.
Torelli Tipo Uno (stolen), Peugeot Nice, Mercier Kilo TT
I have to say.. the torelli tipo uno is really one of the least mentioned FG/SS bike out there but it rode super smooth and I thought it was every bit worth the money. Doesn't come with that much ****ty components.