Picc-Vicc tunnel in Manchester is quite an obvious one, but that obviously needs fleshing out...
Manchester needs something doing with it, but I'm struggling to see what tunnel you'd build.
Pic-Vic will always get suggested, despite the tram having linked the two for around thirty years and the money spaffed on the Ordsall Chord, but what would the link be?
At the moment there are a number of services terminating in Piccadilly from the south/west and two services per hour terminating at Victoria from the north/east.
A "south east to north west" alignment, so that trains would be able to go Airport/Stockport - Piccadilly Underground - Victoria Underground - Salford - Bolton/ Eccles?
That'd be a bit of a "kink" and take services away from the popular Oxford Road/ University areas... would it be limited to "metro" doored stock? Because a lot of the cross city services are fairly long distance without the kind of doors suited to short dwell times at underground stations.
The distance between Piccadilly and Victoria is that awkward middle ground between "short enough not to require an intermediate station" (like a Glasgow Queen Street - Glasgow Central or a Bradford Interchange to Bradford Forster Square link) and "long enough to justify an intermediate station" (like the various London Crossrail proposals). But, if the idea is to link the Airport/Stockport and Bolton/ Eccles corridors (I say *if* but these seem a logical link IMHO) then you'd probably be going east of the Northern Quarter, so an intermediate stop wouldn't be that much use (but then, if you're not going to provide an intermediate station then you're missing out on part of the justification).
The more I think about Manchester, the more I am convinced that any new "cross city" route would be better by tram or tram-train, rather than trying to run it as heavy rail.
I could see the logic in Glasgow - plenty of terminating services, a clear north-south alignment, short enough not to need the expense of an intermediate station, now that most Queen Street services are electric it'd be much easier to work - the only two reasons I'm against a Glasgow Crossrail would be:
1. The problem of getting a north-south alignment when there are two east-west "low level" heavy rail lines and one west-east river plus you'd presumably be going inside the circle of the Subway too - so you'd need to go very deep or be very deep or very creative in how you deal with all of these obstacles
2. Whilst there are a lot of short distance "south side" services (Cathcart Circle, Paisley Canal, East Kilbride etc), there's only the Anniesland service to the north that is fairly short distance (now that the Cumbernauld ones run through to Edinburgh), so there's a danger of importing delays if you start extending long distance services through a tunnel - but this is a secondary issue compared to the first one
Bradford would be a waste of money IMHO, the only justification would be for the kind of people who have an aversion to buffer stops and insist that we fill any apparent gaps in the network - I don't think that there's much demand to justify it - you might as well suggest Epsom Crossrail (to link Epsom Downs to Tottenham Corner) on the grounds that there are two terminal stations therefore there must be some kind of justification for demolishing a golf course to link them. You'd be better investing in Windsor Crossrail (and that's saying something!)