BR trains into London Bridge (normally 8 or 10 carriages) or usually less crowded than the Overground trains (4 carriages); if you are looking to get an overground train from 8am to 8:40am to Canada Water etc its a scrum; BR Trains you may well get a seat. Note though that the London Bridge trains can be unreliable re times (esp the ones that come from Victoria!) and with all the Lon Bridge redevelopment its not uncommon to get caught in "train stacks" for 5/10 mins outside Lon Bridge station.

Going the other way - to Croydon or Victoria, no probs re a seat in the morning.