Sheffield Tramways Company
Summary
In common with many horse tramways in the UK, drivers and conductors wore informal but smart attire — trousers, overcoats, jackets, waistcoats, shirts and ties. Headgear appears to have largely followed the fashion of the day, predominantly the bowler hat, and in the years directly before the corporation take-over (1896), flat caps. No badges of any kind, including licences, were worn on either the jackets or the hats.
It is unclear whether inspectors wore uniforms, or indeed, if the tramway company even employed them; the reasonably large body of surviving photographs certainly show no evidence of them.
For a history of Sheffield's tramways, see: 'Sheffield Corporation Tramways: an illustrated history' by K Gandy; Sheffield City Libraries (1985).
Images
Horse tram drivers and conductors
STCo Horse Car No 33, standing at the Albert Rd terminus in Heeley — photo undated, but taken in 1886 as a poster in the background carries this date, whilst another advertises a race meeting on Good Friday 23rd April, which also occurred on this date in 1886. Photo courtesy of the National Tramway Museum.
A blow-up of the above photo. The two conductors are wearing smart but informal attire; the tall bowler hat with upturned brim being characteristic of the 1880s.
Eades reversible Horse Car No 1, which worked the Moorhead to Nether Edge route, at Nether Edge Depot — photo undated, but given the good condition of the vehicle, probably taken not long after its introduction in 1886. Photo courtesy of the National Tramway Museum. 
A blow-up of the above photo showing the conductor and driver, neither of whom are wearing uniforms.
Another shot of Eades reversible Horse Car No 1, this time in Moncrieff Rd — photo undated, but probably taken in the late 1880s or early 1890s. Photo courtesy of the Tramways and Light Railway Society, with thanks to David Voice.
Horsecar No 44 of 1877 at the Nether Edge terminus — photo undated, but given that at least one of the individuals is wearing a flat cap, probably taken in the mid 1890s. Photo courtesy of the Tramways and Light Railway Society, with thanks to David Voice.
Horse Car No 9, which worked the Sheffield-Attercliffe-Tinsley route, outside Tinsley Depot — photo undated, but believed to have been taken around the time of the Corporation takeover in 1896. The man in the top hat is the Company General Manager, Mr Henry Mallyon. None of those present are wearing uniforms. Photo courtesy of the Tramways and Light Railway Society, with thanks to David Voice.