Yesterday evening I was on a 381 bus from Surrey Quays to Peckham
(which, with hindsight, might not have been my best way to go home, but
never mind that now), and it went over what looked like a railway bridge
at one stage, but no sign of a line. However, a little further along
its route, it passed a very splendid building marked as "Southern
Railway Stables", and there was that about it that looked as though it
had once been next door to a railway line. Anybody know anything?

IINM, the Southern Railway Stables you refer to are alongside the
former Bricklayers' Arms spur on St James's Rd, SE1. Bricklayers' Arms
was the temporary terminal of the South Eastern Railway (and IINM the
Brighton Line) until London Bridge station on the London & Greenwich
was extended and the High Level station constructed. It remained in
use for Goods/Parcels until around the 1970s. As for the stables, I
haven't any idea about why they needed horses!

The place where the line to Bricklayers' Arms branched off can still
be seen as you ride in towards London Bridge on IIRC the Brighton Slow
Line.

Then this morning, I was on a bus down to Streatham and going past the
old LCC Tramways building, now belonging to Arriva after years of being
in private hands. For once, the doors were open, and you could clearly
see the old tram rails marked on the floor. This led me to wonder - did
the trams run further south than this depot, or was that the southbound
terminus on the A23?

You presumably mean the former Telford Av tram depôt. In 1935, the
following routes were running past Telford Av:
8 Victoria - Vauxhall - Stockwell - (Out via Clapham, Balham, In via
Southcroft Rd, Mitcham La, Streatham, Brixton) - Tooting Bway (reverse
direction see 20)
10 City (Southwark Bridge) - Elephant - Kennington Gate - Brixton -
Streatham - Mitcham La - Southcroft Rd - Tooting Bway
16 Emabankment (In via Kennington Rd, Westminster Bridge, Out via
Blackfriars Bridge, Elephant) - Kennington Gate - Brixton - Streatham
- Norbury - Croydon - Purley
18 Embankment (Anticlockwise version of 16) - Kennington Gate -
Brixton - Streatham - Norbury - Croydon - Purley
20 Victoria - Vauxhall - Stockwell - (Out via Brixton, Streatham,
Mitcham La, Southcroft Rd, In via Balham, Clapham) - Tooting Bway
(reverse direction see 8)
22 Embankment (Clockwise) - Kennington Gate - Brixton - Streatham -
Mitcham La - Southcroft Rd - Tooting Bway
24 Embankment (Anticlockwise) - Kennington Gate - Brixton - Streatham
- Mitcham La - Southcroft Rd - Tooting Bway
N2 (Night Tram) Embankment (Clockwise) - Kennington Gate - Brixton -
Streatham - Mitcham La - Southcroft Rd - Tooting Bway

In addition, the following routes terminated at Streatham Library:
2A Embankment (Clockwise) - Stockwell - Clapham - Balham - Tooting
Bway - Southcroft Rd - Mitcham La - Streatham Library
4A Embankment (Anticlockwise) - Stockwell - Clapham - Balham - Tooting
Bway - Southcroft Rd - Mitcham La - Streatham Library