This a new page (Jan 2019) collecting info about taking your bike on the bus in the Sheffield area.
Most buses and Supertram in the Sheffield area do not take non-folding bikes (Folders are generally welcome)
Hulley’s of Baslow however do take bikes. We recommend contacting them to check on your particular journey:- T: 01246 582246 E: firstname.lastname@example.org Tw: @hulleysofbaslow
In this respect we are way behind the USA.
Further afield, I’ve also got wind of informal (but very useful) bike carriage on buses
notably the X70? (connexxions?) Wetherby-Harrogate and around Tadcaster
(possibly 843 Leeds-Malton Coastliner).
Borders Buses is buying 3 new buses, specifically equipped to carry bikes Edinburgh-Peebles-Galashiels, and there is a further project.
Update from Dave Holladay, April 2019.
1) Trent Barton’s Red Arrow (Nottingham-Derby-Chesterfield coach services) were carrying bikes with bike securing racks in the hold
2) A useful cross reference might be put into the rail section for the Hulleys routes that follow the Hope Valley should you get stranded.
3) X70 is the only service I’ve heard regular reports of informal carriage – Coastliner would be very useful though. Elsewhere the 599 (every 20 min in summer) open top service along A591 by Windermere carries bikes on lower deck, & Scottish Citylink is preparing to deliver some useful changes for cyclists with their new app & booking systems – especially useful are the 976 Glasgow-Oban (when the 08.21 train is fully booked catch 08.30 coach) the 915 (Glasgow-Uig) and the 961 (Ullapool-Inverness) basically avoiding some big hills & less pleasant roads from key ferry crossings.
4) Went to a meeting in Manchester, with a report on Mobike, and a discussion on how TfGM will deliver a new bike share scheme early in 2020 with access to bikes across all TfGM area. It would make great sense to have any schemes for Sheffield, Leeds, Wakefield, Liverpool &c seamlessly accessible via a single user ‘account’ even if each area had different system suppliers.
5) A real bargain is the Peak Wayfarer unlimited rail & bus (& tram?) across a swathe from Wigan to Grindleford, the only snag being that the train you get from Sheffield has to stop at Grindleford to use a Sheffield-Grindleford ticket to bridge the gap.