Durham and Newcastle because Newcastle think they're the best university in the north of England.
Uni of Nottingham and Nottingham Trent, its mostly banter though but some people take it seriously
London met vs Harvard University
In London: UCL vs KCL aka Godless Scums from Gower Street vs Strand Poly Chavs
There are stronger name calling, but I shall refrain or I would get told off by moderators
i always thought Birmingham vs Warwick idk
Oxford v Cambridge
UCL v KCL
UCL v Imperial
UCL v LSE
LSE v LBS
LSE v Imperial
Edinburgh v Glasgow
Edinburgh v St Andrews
Leeds v Sheffield
Nottingham v Trent
UCL v Durham
KCL v Warwick
Durham v Warwick
ABRSM v Trinity College London
Cardiff v Swansea
Warwick v Manchester
Birkbeck v Open U
London v Oxbridge
UCL and KCL have been rivalries for several decades.
Any red brick and poly which are in the same city/area
It depends on the subject surely like Warwick vs Imperial when it comes to Maths, or UCL vs Warwick vs LSE when it comes to Economics or UCL vs QMUL when it comes to medicine, or Imperial vs Oxford for Engineering i could go on and on...
Newcastle and Northumbria
United by geography, divided by job prospects.