I flew the bike to Vancouver, then returned fro Toronto, in 2002 so prices will have changed in the intervening years but from memory it was about £1500 return for the bike plus my own return ticket. The bike sat in a simple aluminium box and went in the hold of the plane.
My 2006 trip - with Wallenius Wilhelmsen, Southampton to Newark and back - cost £547 single - plus my own flight ticket. I would have liked to have sailed with the bike but that wasn't permitted.
No carnet required for the US. Incidentally flying a bike into Canada in 2002 was appreciably cheaper than landing in the US. I don't know the reason for that.
NOTE - a visa is required if your stay in the US is 90 DAYS OR LONGER. I almost got caught out with that one as my last trip last 92 days.