Best bet is what MrsRichUK & Alistair have described, to avoid any difficulties with the US Dept of Homeland Security upon entry to the USA - which I'm all for smooth traveling!
I'm dual American & British. When flying to the USA, I:
1) Check in at UK airport using US passport
2) Enter US using US passport
3) Check in at US airport using US passport
4) Enter UK using UK passport
It's not just the question of dual nationality & what you can/can't do - it's to do with the Advanced Passenger Information (API) system where flying into the US, the passport that you use to enter US immigration should match the one on the airline passenger records manifest. Otherwise, you could
potentially run into problems with Homeland Security. I suppose the UK could end up requiring that too if it wants to be more awkward about things (making this problematic for us duals) but AFAIK the US is far more sticky about it at the mo' than the UK is.
That way I'll be on the passenger list as American upon my arrival in the US which should cause less headaches on the Advanced Passenger Info thing.
Yes! Cross-posted. (Traveling to/from the US, I always carry both passports, should I ever be in the situation that I have to produce either one.)
Also if you're not an American citizen (showing your US passport upon entry to the US), Homeland Security will be looking for your visa to enter the country, or that you have pre-registered online for the Visa Waiver Program & any required Visa Waiver form if you're from one of those countries.