I have tried a few different folding bikes, but I haven't tried them all, so this is not a consumer report. I have a prejudice against the folding ones with little wheels, though I began my music touring with a small folding Peugeot with little heels and it held up for many years. I now mainly use full-sized folding bikes that feature some form of suspension---either in the front fork or the seat. I once go really sore wrists and I think it was from riding a lot on cobblestones in SoHo without suspension. For a guitar player this was worrying. Montague and Dahon both make folding bikes with full-sized wheels. Brompton, Birdy, Moulton, and Dahon all make folding bikes with small wheels.
The full-sized bikes fold into a large suitcase with wheels (that also has room for a helmet), which can be checked as a second piece of luggage. Once upon a time you were allowed two pieces of luggage with no extra charge---that's rare now. I've gotten stuck with $125 sports-equipment charges (I think this charge was meant for skis or golf bags), so I'd think twice about throwing in the bike for a short trip these days. If I'm going to be somewhere for the week it is not only practical and fun but also economical, even with the baggage charge.
An alternative to all this luggage and packing is to rent a bike when you get to where you're going. It's getting easier to do this locally. I recently rented a bike in berlin for a week and in Salvador, Brazil, for two days.