Can you include a signature in a bitcoin payment? I know you can use the bitcoin client to sign messages (for which you own the private key) and verify messages from other public addresses. But can you actually embed a signed message into a transaction so its on the blockchain for anyone to verify? When I say "signed message" i'm referring to an arbitrary message like in the attached example.
No. This is not possible, although, technically it could be possible by adding n child transaction each formed as a tx out of you original transaction. The child transaction then could each have a OP_RETURN (data only) output. With a OP_RETURN output you can store up to 80 bytes (mind 5286). A DER signature will be around 72 Bytes (71.5 in avg)

... so not much to left besides the signature (~8 bytes per OP_RETURN tx as tx out of the original tx). So maybe use OP_RETURN txout 0 to n-1 for the text/data and n for the signature. Obviously this will increase your transaction fee a little because you need at least 2 additional transactions.

Conclusion:
Not possible unless your are happy to experiment with forging your own custom transactions.