This paper describes a two-phase study conducted to determine optimal target sizes for one-handed thumb use of mobile handheld devices equipped with a touch-sensitive screen. Similar studies have provided recommendations for target sizes when using a mobile device with two hands plus a stylus, and interacting with a desktop-sized display with an index finger, but never for thumbs when holding a small device in a single hand. The first phase explored the required target size for single-target (discrete) pointing tasks, such as activating buttons, radio buttons or checkboxes. The second phase investigated optimal sizes for widgets used for tasks that involve a sequence of taps (serial), such as text entry. Since holding a device in one hand constrains thumb movement, we varied target positions to determine if performance depended on screen location. The results showed that while speed generally improved as targets grew, there were no significant differences in error rate between target sizes =9.6 mm in discrete tasks and targets =7.7 mm in serial tasks. Along with subjective ratings and the findings on hit response variability, we found that target size of 9.2 mm for discrete tasks and targets of 9.6 mm for serial tasks should be sufficiently large for one-handed thumb use on touchscreen-based handhelds without degrading performance and preference.