Most of the replies so far have referred to EastCoat services (which can be seen, as others have posted, on a rather unpredictable basis).
But the OP's question didn't seem to be Operator-specific.
There are many regular services over the High Level bridge - all the Newcastle - Sunderland (then Durham coast southwards) services go that way, and I am often caught out by the occasional TPE southbound service (I'm not able to tell you their destination) which departs to the east (northbound in Newcastle) but then traverses the High Level Bridge to gain the southbound ECML.
It does say EC in the thread title!
In my experience this is most commonly used by the Newcastle to Durham Coast services, as you say. It is also used by a few XC services, particularly I think those which will be reversing out of Birmingham New Street (so I guess those heading to Oxford and beyond). If you're looking for HSTs then I
think I've been over it in the XC HST which tends to be in the north-east on Sunday mornings, but that might be a faulty memory. Quite a few of the Hexham and beyond trains use it, as they often depart from the Northbound only platform 1 from Newcastle - I think this includes the Scotrail services to Glasgow. I've also seen the occasional TPE on it, as you point out. Grand Central trains can often be seen round there as well, I presume going to and from the Heaton depot?
The Central in Gateshead, a 'Head of Steam'-owned pub has a balcony which overlooks the track between the High Level Bridge, though you tend to only see Northern Rail and the occasional Grand Central heading to and from Heaton on it. If you crane your neck round you can see the trains which are heading to/from the High Level Bridge as well!