Are there any particular trails to do, or is it worth scurrying over the border to Italy on a lift, or round the corner to Val?
Better places to go for that kind of riding TBH. You can't cross the Italian border from Tignes (it's not even particularly close).
Tignes isnt particularly pretty in summer. I went there last year and had a brilliant time. Free lift pass, loads of trails - mostly big DH stuff. Wonderbois is a fantastic trail.
Andy, the trail down to Les Brevieres is rather good singletrack. It takes a while to get good though. Trails arent well sign posted and a lot are similar. Its also susceptible to high winds, so chairlifts are often closed (we only managed to ride 1 side of the mountain).
Val has better trails imo, but there arent many. 3 or 4 I think. They are nice and long though. Heres 1 of my vids:
Lifts have been free for the last couple few years though.
St Foy, Les Rosiere/La Thuile, La Plagne + Les Arcs just down the road.