Is it with their left hand, using its thumb to tap UI elements? Or do they use their right hand like right-handed people? Is there any research into this matter with published numbers?
I got inspired, did some digging in my search engine of choice and found a very recent study by Henry Ford Hospital (published in February 2012) that gives a more or less clear answer on the debate:
People hold phones in the hand that is opposite to the dominant side of the brain (where speech & hearing centers are located).
This means that the majority (70%) of right-handed people do indeed hold it with their right hand and left-handed people with their left hand because their speech centers are in the left and right hemispheres respectively.
However, there're still 30% of people who use the non-dominant hand for phones (like me) and people who are supposedly ambidextrous.
I am left handed, and I do use my thumb to tap UI elements wile holding a smartphone in my right hand.
It seems as though left handed people tend to hold a smartphone with their right hand, as holding a smartphone does not require a lot of dexterity. Also, smartphones are usually optimized for right handed people, as the majority of humans are, in fact, right handed.
However, it all comes down to a person's preference. For example, it's like with computer mice - most left handed people will use their right hand to control the mouse, because almost all mice are ergonomically designed to fit someones right hand. Now, computer mice are being created for left handed people to use, but are not very popular because most of us lefties have grown up moving the mouse with our right hand and typing with our left hand. If I had to type with my right hand and move the mouse with my left hand, I wouldn't be nearly as efficient. It would be like learning to use a computer all over again.