Bulwell, Carlton, Attenborough are served mostly ever hour and Netherfield is served irregularly.
Bulwell is served by the tram, which has the advantage of serving Nottingham city centre so is probably more attractive for local journeys. I believe the reason Bulwell only has hourly stops is because somewhere had to take the hit when the RHL was singled for the tram to be built, and being so close to the city it was felt it would suffer the least by only having hourly heavy rail.
Attenborough is not, I think, that large a traffic generator so probably doesn't justify more frequent calls on what is a fairly heavily trafficked two-track railway (by provincial standards. I know it's almost empty by Southern Region standards!). Netherfield I don't know about, although it is very close to Carlton so probably isn't felt to be a traffic generator towards Nottingham.
I do personally feel that Carlton should have two trains an hour,
but the current service patterns mean that they wouldn't be half-hourly so probably wouldn't be wonderfully attractive.
Many parts of the city such as Arnold, Mapperley and West Bridgeford are not served by and rail services.
Those parts of Nottingham have no rail lines which could have any rail services running on them, so they're not going to be served by passenger trains...
Syston which is served hourly and South Wigston which is served two hourly.
Syston is awkward because it's on a single track section and it's location means that any train serving it blocks Syston South curve because of the signalling layout. I'm not totally sure there is justification for more frequent service off-peak for Syston or any of the other Ivanhoe stations, nor am I sure how additional peak journeys could be introduced.
Spondon and Peartree both of which have a very limited service.
It's ten years since I last woked trains that way, but I seem to recall that Spondon was almost purely an industrial area with the residential area being some distance away and well served by local buses. Given that even then the factories seemed to be closing down, and assuming they haven't been replaced with new residential build, and that it's on a fairly congested two-track section of line I don't see any justification for an increased service at Spondon.
Peartree was I believe intitally provided with a regular service but loadings were always disappointing. I don't know the current state of the area, but given again that (as with Spondon and Attenborough) that it's on fairly intensely used two-track and it would swallow paths approaching Derby, is there any justification for increasing calls beyond "Werrrllll, it'd be nice to have?"