Sheffield had two periods. The first five years of de-regulation had lots of "breadvans" and other second hand vehicles pressed into service on random routes by any old cowboy - similar to many large cities - but SYT (South Yorkshire Transport - later Mainline, now First) bought a number of them and things calmed down.
But nature abhors a vacuum, and competition came back (just as SYT were going through a court case for abusing their monopoly position). The "worst" era was around 1993/1994/1995 when Sheffield Omnibus competed on pretty much every route they could - it's ridiculous nowadays when you look back and remember that there was an Omnibus service every ten minutes competing with a Mainline service every ten/fifteen minutes on corridors that now only have one bus per hour in total (e.g. the 93/94 from the city centre to Meadowhall via Grimesthorpe, the 17/18 from the city centre to Fox Hill via Hillsborough, the 1/11/21/57/66/67 from the city centre to Stocksbridge via Hillsborough).
Omnibus even ran competing evening services on the 52 (where Yorkshire Terrier were the "daytime" competition to Mainline) - I know that some "newcomers" provide an all-day service when competing with an established operator but I'm not aware of any examples of a "newcomer" providing a commercial evening service in competition with an established operator when that "newcomer" don't even provide the daytime service.
Omnibus had the clever tactic of buying lots of redundant double deckers from Preston, which were coincidently in the traditional navy/cream colours of old Sheffield buses - they were pretty good at identifying gaps in the Mainline network (and sections of route where the incumbent operator went a bit round the houses rather than running direct, or where Mainline served the quieter side of the city centre around the Interchange so Omnibus could run up High Street and be a lot more convenient for passengers).
In those days, the *weekly* ticket price went down to two quid at one point (not single ticket, not daily ticket, I'm talking seven days of travel!), as the two operators fought tooth and nail (the Mainline ticket - I think it was called "red saver" - was only valid on the generally north-south corridor that Omnibus operated on - but still very generous if you lived on those sides of town). There were duplicates, there were unregistered services, there were often spare buses sat at places like Lane Top waiting for the "other" operator's bus to turn up so that they could ensure that they were first to the lucrative stops ahead (e.g. Firth Park, in that example).
Whilst there were other independent operators in Sheffield, nothing got as bad as the Omnibus/Mainline days (most of the post 1990 ones stuck to certain key corridors - e.g. the two main east-west Stagecoach routes - the 52 and 120 - have been operated by their predecessors for thirty years now). Eventually, Omnibus was bought by Yorkshire Traction, along with Andrews, Yorkshire Terrier and South Riding and things became fairly stable - a "death of a thousand cuts" as Tracky tried to rationalise the assortment of routes they'd inherited (culminating in the "Lowedges to Wath via High Green" service 72!) and focus on a few corridors. Also, by the mid '90s, the new-fangled Supertram was taking a huge chunk out of some key corridors, so the days of "bus competition" were replaced by "bus survival", as operators tried to keep viable services.
Wish I'd had a digital camera back then - there were some horribly congested bus stops, superannuated "sheds", mayhem... If I were younger today, I don't think that I'd have got as interested in buses - so bland in comparison.
arriving on the X50/X70 from Darlington!!! 
(on a finance agreement) that competed with Busways. 
