The Mosborough Expressway (south east of Sheffield) and the A61 (north of Sheffield)
For those unaware with this neck of the woods, Sheffield city centre was the junction between the north-south A61 (North Yorkshire - East Midlands) and the east-west A57 (Lincolnshire - Merseyside)
Sheffield is close to the motorway network, with a duel carriageway between the city centre and the M1 (the A57, Sheffield Parkway, dumping people at a large roundabout, like a northern version of the M32 between the M4 and central Bristol)
However, whilst it looks good on paper (Sheffield city centre has a reasonably fast road to a reasonably close part of the motorway network - albeit the roads between the city centre and the nearest bit of motorway at J34 can be horrible because you need to deal with the traffic for Meadowhall/ Arena/ IKEA etc), it's more a case of "Sheffield has one fastish road from the city centre to the motorway" rather than "Sheffield has fast links from the city centre towards the way that people would be travelling"
SO, anyone heading from central Sheffield to Nottingham/ Birmingham/ London needs to travel north east on the A57 to reach the M1 at J33 before travelling due east towards Thurcroft and only turning south once you reach the large triangular junction with the M18
Anyone heading from central Sheffield to Leeds/ north needs to travel north east on the A57 to reach the M1 at J33 before travelling north west towards Barnsley
This means that a lot of traffic uses local roads, like the Mosborough Parkway to get towards J31 Aston (at least half a dozen roundabouts with no grade separation) or via Meadowhead to get towards the A617 at Chesterfield towards J29 (traffic through Woodseats is horrific, as is the roundabout at Meadowhead) or the A61 to get towards J36 at Tankersley (long single carriageway sections, busy roundabout where you cross the eastern end of the "Stocksbridge bypass")
Whilst the Sheffield Parkway is a useful way of getting to suburban Rotherham and also for the M18 towards Doncaster/ Humberside, it'd have been a lot better for Sheffield if we'd replaced the one "fast" connection to the motorway for more of a "loop" off the M1 around J30 past the new "townships" around Mosborough/ Crystal Peaks, past the city centre and then join up with the motorway network again around J36 - maybe it'd head from J30 towards Gleadless (where Peter Stringfellow arranged for The Beatles to play a gig many years ago), down the Gleadless valley towards Meedsbrook?
There were plans for a proper "Mosborough Expressway" (rather than just the "Mosborough Parkway" that has a lot of roundabouts) and some ambitions for a proper fast road through northern Sheffield too, but never came to much, and our local roads are left to struggle with the vehicles that try for a more direct route than the inadequate route that the Sheffield Parkway takes. Maybe things would have been different if the fast road north had followed the M18 alignment to the A1 near Doncaster, rather than forcing the M1 over the viaduct at Tinsley - but, just like with plans for "High Speed" railways, Sheffield is stuck between "box ticking" and "something genuinely useful for Sheffield"
Sheffield has always had strange relationships to roads though - instead of a proper "Ring Road" we built fast dual carriageways through the centre of the city (Arundel Gate etc), taking all traffic through the middle of Sheffield at street level, splitting the city centre up, taking a long time to rectify!
(a beter faster road over the Pennines towards Manchester would be great too but a lot harder to build than designing a road through the "blank sheet of paper" of the New Town of housing estates around what is now Crystal Peaks)