I'm looking to rent an apartment (or something) for 3-4 weeks in March (as an exploratory recce) in the torrevieja area: I'll have a car. Any recommendations about which specific residential areas might be preferable? Any places to avoid?

The centre of Torre can be a bit rough and parking will be difficult. You might want to avoid the centre or search in one of the many urbanisations outside the town. Google maps is your friend. Enjoy your visit.

By rough I guess you mean Spanish?

Not at all. I mean rough as in like many city centres in any country it is overcrowded and attracts criminals and undesirables of all nationalities.

Well it is true it is overcrowded in the summer months and true that parking is an all year problem. As for crime , you are more likely to be burgled in areas with high foreigner numbers and empty properties but these are not in Torrevieja

To be honest (having lived there) I would avoid the whole area. It is overbuilt, poorly planned and managed, scruffy beaches, and dying central commercial area (due to La Zenia boulevard shopping centre) and generally not the best of what Spain has to offer. I would definitely avoid the Parque del Molino area unless you want to watch the working girls looking for business.

But hey each to his own so go and have a look and if you love it fine. But if you own a car you will need a guaranteed parking space otherwise be prepared to drive around forever in the summer (when it is totally packed out) looking for a space (been there done it!). Good luck.

There are many crimes worse than burglary :-(

According to data it has a high percentage of foreigners? Regardless of that it will depend what you are used to and it's always good to visit at different times. Visiting in March will be very different to July or August.
In general the place seems to be very unspectacular. You have a lot of concrete, it's flat and boring. If you've driven through Spain, you'll probably find many places that have more to offer. But for some it's probably paradise.

According to the 2018 stats out of a population of 82599 - 32033 are foreigners. This is of course based on the padron and so excludes those of any flavour who are not registered.

Datos del padrón de Torrevieja - Diputación de Alicante

I would say there are a lot of empty properties out of summer/fiestas times.

Fair enough, each to their own but I feel Torrevieja gets a bit of a put down by foreigners. It might look less than ideal but that is because it is a working town with most Spanish living in the centre. Its problems result from huge summer residents who stretch the towns structures. This coupled by many foreigners not on the padron means funding ain't great. Having said that it's very cheap to live there. Much cheaper to rent and buy than the Chan traps in Punta Prima etc.People are also friendly and even El Molini has a friendly community even if they come from the poorer social side of life. Only other thing need to say was the one fight I saw here was between Swedish and English football fan on the paseo.

Yes 1sexsmith - that is fair comment I lived there 2000-2002 then returned in 2008 (I was running a business and was more convenient to live in town) moved out 2014. I had a lot of fun but life moves on and I found somewhere that suited me better.

I actually wouldn't say it's put down by foreigners and is probably the opposite. Most Spanish I know say they dislike the place and the high amount of foreigners show that certain nationalities love the place:
British & Russians are at the top followed by Ukranians and Moroccans. I just think that there are various other less known places where you can find a decent beach, promenade and a bit of variety + cheap too.

I think that some people refer to torrevieja as places outside of the town . I know many Brits who talk of playa flemenca as torrevieja. Personally I prefer in town to living in somewhere where English is immediately used all the time. Was in La Zenia a while ago and every shop insisted on speaking English even though I spoke Spanish!!

Had the same when I lived in Holland>

"Ik spreek geen Engels" worked for me every time