Sheffield:
We had quite a few missing for a number of years, before the "partnership" between First/Stagecoach saw a number of services renumbered and the 1/2/3/4 becoming the services from the City Centre to the Northern General Hospital (to replace routes with much higher numbers)
0. No current service, but we did have "Bus Zero" in the 1990 (Mainline's innovative attempt to compete with taxis in the Student areas, complete with condom machine on board and a route map that only showed pubs/nightclubs with nothing as boring as actual street names!)
1. Currently one of the major cross city routes (Stagecoach's High Green - Batemoor, with First operating a complimenting 1a over half the route) - in older days it was a less frequent service to Longley, always seemed a bit of a strange route to have the number 1
2. Sheffield - Barnsley (the old 165/265 corridor). Probably better known as the number for the long running "outer circle"
3. No. Was a ten minute service before being chopped into the 95 (northern end) and 56 (southern end)
4. The service from Sheffield to the big XPO "shed" near Grimethorpe, I think the only local bus service operating on Christmas Day - only a few journeys a day, to coincide with shift times, but also a handy way of linking Holyand etc with Sheffield (the bus services like the 271 dwindled away over the years, given the train competition, until the XPO place opened and enough people were "commuting" in the other direction to sustain a bus service (I don't know if it's subsidised though)
5. A fairly minor tendered route (City - Grimesthorpe - Firth Park) - Grimesthorpe is not to be confused with the old mining town of Grimethorpe by Barnsley where the 4 goes near. The 5 was the Sunday service to the Strines and other parts of the Peak District in older days
6. Once the Darnall - City replacement for the old 70/71... then became Darnall - City - Fulwood and then Darnall - City Millhouses, now just City - Millhouses (i.e. one of those routes which has continued in operation for decades but has completely changed the route and no longer serves any of the roads it used to)
7. Now a cross-city Stagecoach service (Ecclesfield - Crystal Peaks), once the "other" route that replaced the 70/71 circle
8. First's Ecclesfield - Crystal Peaks service that used to be co-ordinated with the Stagecoach route (until Covid cuts saw both companies tinkering around with their timetables and leaving gaps). For many years the 8 was the "inner circle", albeit one that ran through the city centre
9. Now the Darnall - City replacement for the 6 (i.e. running over the route that the original 6 ran), once the other version of the "inner circle" (although the number 8 was retained for longer)
Until the "partnership" of the last decade or so, we've not had many "frequent" single digit services, most of the higher frequency corridors had numbers between 51-100 - I'm not sure why - even the independent companies who competed with SYT/Mainline either chose services numbers that were identical to or similar to the routes they shadowed or (in the case of sut and Sheafline) went for three digit numbers
The 9 also had an older route which I don't know much about other than it ran between Torphin and Newhaven, and then a more recent one as a summer only service running between the Botanics and National Museum of Scotland in 2014 (I think?).
9 was Torphin to Granton, then Torphin to Newhaven. After disappearing in 1989 the service number was then recycled twice in quick succession, first from 1991 to 1994 for one of LRT's West Lothian services, then from 1995 to 2000 as a variant of service 8. Its most recent use was for the aforementioned summer-only service.
@DunsBus beat me to it - the 9 was similar to the 10 other than it ran via the Shore at Leith (instead of Great Junction Street - back in the days when that part of the Port was fairly quiet, before the Civil Service relocation and the 22 running every few minutes) and then later I think that the difference between the 8 and 9 was that one served Polton Mill whilst the other served Bonnyrigg (in the days before the 31 was split from the 33 and extended into Midlothian instead - I think that a lot of northbound journeys were "9A" to show that the service ran via Silverknowes beach after passing Muirhouse)
(Easter Scottish operated a C9 in the 1980s but that's outside the scope of this thread since it had a prefix)
Perth does not have a 8 at present, don’t know about the past.
I think that the 7/8 was the cross-city service, before being simplified into one route(?)
(or cross-town service, depending on what you classed Perth as at the time)
It's not quite a city/town (although it does have a population well over 50,000), but there is no route 8 in Northern Ireland's national Ulsterbus network and I don't think there has ever been one
That's interesting, given that everything is centrally controlled (and it's not an "unlucky" number like 13 - in fact 8 is quite lucky in some countries)