The instruments on the Tropical Rainfall Measuring Mission (TRMM) satellite have been observing storms as well as rainfall since December 1997. This paper shows the results of a systematic search through seven full years of the TRMM database to find indicators of uncommonly intense storms. These include strong (> 40 dBZ) radar echoes extending to great heights, high lightning flash rates, and very low brightness temperatures at 37 and 85 GHz. These are used as proxy variables, indicating powerful convective updrafts. The main physical principles supporting this assertion involve the effects of such updrafts in producing and lofting large ice particles high into the storm, where TRMM’s radar easily detects them near storm top. TRMM’s passive microwave radiometer detects the large integrated ice water path as very low brightness temperatures, while high lightning flash rates are a physically related but instrumentally independent indicator. The geographical locations of these very intense convective storms demonstrate strong regional preferences for certain land areas while they are extremely rare over tropical oceans. Favored locations include the south-central United States, southeast South America, and equatorial Africa. Other regions have extreme storms mainly in specific seasons, such as the Sahel, the Indian subcontinent, and northern Australia. Because intense storms are distributed quite differently from rainfall, these maps provide some new metrics for global models, if they are to simulate the type of convection as a component of our climate system.