1. a simple shallow bottle trap underneath the sink to make use of the
cupboard space http://tinyurl.com/jubtf

Building regulations require for a kitchen sink a minimum diameter of 40mm
and a seal depth of 75mm. This trap has the diameter, but only 38mm seal.
However, you could argue that your downstream trap meets your requirements,
provided it has a 75mm seal.

http://www.screwfix.com/app/sfd/cat/...08026&id=48341

Has a 76mm seal and would meet building regs even without a downstream trap.

Would you actually need the trap below the sink if I have a running
trap further down the run?

Technically not, but it would stop dishwasher/washing machine smells coming
out of the sink.

Personally, I don't bother with all that. Washing machine/dishwasher get
their own traps with standpipes. Sink gets its own trap. Wastes are T'ed
together after the traps with no shared traps until you get underground.

http://www.screwfix.com/app/sfd/cat/...08026&id=18640

With this method you don't get one appliance's smells coming out of the
other.

Christian.