This is a topic I expect to expand a lot on, so consider this an early introduction only. First rule: As a foreigner, you don't want to live anywhere outside a 10 km radius of the city centre. I'm not a snob, but trust me, I don't expect that you will enjoy your experience if you do. Don't get me wrong, the Luxembourgers are lovely people, but you'll find it hard to integrate unless you live close to the buzzing, international city of Luxembourg.
Luxembourg City is divided into several "quartiers" or areas, the main ones being (I'm actually going to try to cough up all of them, just don't expect me to do it in alphabetical order): Belair, Cessange, Dommeldange, Beggen, Muelenbach, Bonnevoie, Gasperich, Belair, Gare, Centre, Neudorf, Kirchberg, Weimerskirch, Limpertsberg, Weimershof, Cents, Merl, Rollingergrund, Eich, Clausen, Grund, Kohlenberg, Hollerich, Hamm, Puelvermuhle, Pfaffenthal, Verlorenkost and I think that's it. The biggest areas are Bonnevoie and Limpertsberg.
The "best" areas are supposedly Limpertsberg and Belair, probably in that order. But bear in mind the following: Limpertsberg being the second largest (to my knowledge) has both great and not so great areas within it. Parts of it feel really remote, have very little amenities and are a significant walk from the centre. So when looking for a house or an apartment in Limpertsberg, pay close attention to the area. Being on a hill, it's relatively quiet in terms of traffic, apart from the places around the main arteries. Architecturally speaking, it's a funny mix of nice townhouses and awful apartments from the sixties. (However much I love Luxembourg, I have to make it quite clear right now that I consider Luxembourgers to have their taste in their arse when it comes to architecture and interior design. That's the danger of growing too rich too quick.) Belair is similar, but better located. Houses are extremely expensive in this area. Keep in mind that when Luxembourgers and Luxembourg residents speak of good and bad areas, it's all relative. Frankly, there are no dodgy areas in Luxembourg. Ok, you have a few tramps and druggies around the station (Gare), Hollerich and a limited area of Bonnevoie, but in most other countries the ones you do see would be no worse than your regular dinner guests. Nothing to worry about. Bonnevoie is the largest area, and probably the most exciting. No, I don't live there myself, but I recommend it to you. It has a hugely interesting mix of people and lots of shops and restaurants. For Luxembourg, it's still reasonably priced, but trust me, that will change. The area is being done up so the area is improving quickly. Clausen and Grund are probably the most picturesque areas of town - and benefit from being bang smack in the middle of everything - but the noise in the evening will be unbearable to all but the most hardened madrileños. Never mind trying to find a parking place after work. Gasperich is nice, varied, a bit away from it all, but with a nice village feel. Lots of former social housing turned young professional and families on a budget. Kohlenberg is basically the posh area of Gasperich, and boils down to what is pretty much just one street on a hill. Not really a rental market, but caters to house buyers who have significant budgets and wish to live in large, detached houses within the city limits. Verlorenkost is to Bonnevoie what Kohlenberg is to Gasperich - the posh bit. Luxembourg Cents is a great area for families with great sports facilities and a green, quiet environment. Its proximity to the airport makes it both convenient and perhaps a tiny bit noisy in some parts. However, one to be recommended. Adjacent Neudorf is less upmarket, but up and coming due to its great location right next to Kirchberg - home to the European Institutions and a lot of the banks. Its significantly Portuguese community is being slowly replaced by a very international crowd who are doing up the old houses, and a lot of homes have recently been torn down to make room for smaller apartment blocks. It's in a valley, so avoid the right hand side when coming from the city centre, as it gets very little sunlight, especially in winter. The farther away from the city you come, the more it clears up. Forget what I said earlier on about Limpertsberg and Belair being the best areas. The king of the hill is Weimershof, a small residential area just next to Kirchberg. This place is great, but very expensive and has virtually no shops and few restaurants. Kirchberg itself is one of the most busy business areas during the day, but feels slightly sterile from a residential point of view. Cessange is quite far away from the city centre and although it feels nice and landly, it suffers a bit from aircraft noise as it's under the flying path. Which brings me nicely on to Hamm. Hamm would be a lovely place to leave if it weren't for the fact that the wheels of landing planes almost touch the rooftops. This makes it into what is probably the cheapest place to live in town. On the upside, it's right next to the large Itziger forest with mountainbike and running paths aplenty. A very pretty area. Beggen is probably to be avoided due to heavy traffic and constant roadwork. However, it does have a lot of shops and amenities, including some great Asian restaurants. Dommeldange, which is next to Beggen has a cosy village feel, but is a bit of a hike from the centre. And if you're driving through and the barriers come down at the level crossing, you might as well turn around as you could be stuck for a good 10 minutes waiting for the train to pass. I'll write more about these and the other areas later.