Found the reason for constant punctures (I think), but...

[David Marshall]

Here is my guess as to the bump at the white spokes.  The valve is probably between the white spokes.  Let down the tire and check that the valve stem can be pushed in until it hits the outer tire surface and is loose.  To do this you will have to loosen the round valve stem/rim locknut (throw it away!).  Pump up and the hop should be gone.  If not it may be a hop in the rim and need to be trued.

[Rachmaninoff]

Could also be old tyres, with the casing shedding bits of metal?

I certainly hope not! Paid an arm and a leg for tyres not too long ago...

[Rachmaninoff]

Thanks for all the advice, guys! Good to know there's a heap of helpful blokes on here.

[Rachmaninoff]

Here is my guess as to the bump at the white spokes.  The valve is probably between the white spokes.  Let down the tire and check that the valve stem can be pushed in until it hits the outer tire surface and is loose.  To do this you will have to loosen the round valve stem/rim locknut (throw it away!).  Pump up and the hop should be gone.  If not it may be a hop in the rim and need to be trued.

Off the top of my head I think the valve is actually directly opposite the white spokes, but will have a gander tonight just to make doubly sure.