I arrived here last month for work and have just been looking for apartments.
On a budget of around 1,000-1,200 Euros per month for rent I saw some horrible places and also some very nice places. For that price-range though, and for someone living alone you should be able to find somewhere close-ish to the centre, but if you can push your budget a little more you will find somewhere nicer maybe.
Be warned though that space is going to be an issue. For a single person that is fine and you can find a lot of studio apartments. My girlfriend is moving over in April so our place needs to be big enough for two and while the place I am in the process of choosing is reasonable - for Paris - at 35 sqm it is far from spacious.
If you take the centre as being the area around Chatelet-Les-Halles and Hotel De Ville Metro stations, then there will be options for you in the livelier areas around there 3rd, 4th (North of Seine) and 5th, 6th (South of Seine) but they will be most expensive. Moving a little further out, 10th, 11th, 12th, 14th, 15th it will be a little cheaper and maybe a little bigger but not as close to the 'life and soul' of Paris.
However, what I have found is that while there is a 'centre' each arrondissement has its own life, like lots of little villages/towns really, so you should be able to find entertainment and night life in most areas - although some are a lot quieter and more geared for families.
The one problem I have found is that there is no hard and fast rule on how much space you get per Euro, thus you could find somewhere 50 sqm one day and the next day be shown somewhere 30 sqm for the same price. It can be frustrating and even though I really like the apartment I am hoping to move into I am still left thinking if there was somewhere bigger and better that I could have found had I kept searching.
I currently live in the 3rd in a company apartment but will be moving to the 12th if my deal goes through. It is probably a 30 minute walk from the centre but 8-10 mins by Metro. Paris is a very small city by comparison to some so wherever you are in the relatively central area you should only ever be 20-25 mins on the Metro from the heart of the city.
Hope that info is useful. Any questions give me a shout. It can be daunting at times but there is a high turnover of property for rent so you should be able to find something suitable.
-- 26 March 2011 --
Sorry, I read you not having kids line as being that you were single, obviously if you are coming with a partner bear in mind the size/space issue! I am looking at 35 sqm for 2 people and even that seems very cramped. However it all depends on the layout. One place I saw was listed as 40 sqm but actually seemed more cramped than another I saw which was only 32 sqm...
As for other costs I have been told the deals on TV/Internet/Phone are all roughly around 35-40 Euros a month. I have found groceries quite costly though as there are few 'super' supermarkets where you would usually be able to buy in bulk and get 2-for-1 deals etc... Mostly they are mini-markets where costs seem high to me and variety limited. It will take a bit of shopping around to find the best places/prices.
And drinking out I find very expensive, but that is to be expected of a big city.
-- 26 March 2011 --
Oh and one last thing, this place should make you feel at home. Also has great views of Notre Dame from the front terrace...
[EDIT: Apparently I don't have permission to post links, so... Google 'The Great Canadian Pub Paris']