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.