The fine-scale structure of global (30N-30S) tropical rainfall is characterised using 13 years (1998-2010) of 3-hourly and daily, 0.25-degree Tropical Rainfall Measuring Mission (TRMM) 3B42 rainfall product. At the outset, the dominant timescales present in rainfall are identified. Specifically, the Fourier spectrum (in time) is estimated in two ways (a) spectrum of spatially averaged (SoSA) rainfall; and (b) spatial average of the spectrum (SAoS) of rainfall at each grid point. This procedure is applied on rainfall at the 3-hourly and daily temporal resolutions. Both estimates of the spectrum show the presence of a very strong seasonal cycle. But, at subseasonal timescales, the two methods of estimating spectrum show a marked difference in daily rainfall. Specifically, with SoSA the variability peaks at a subseasonal timescale of around 5 days, with a possible secondary peak around 30-40 days (mostly in the southern tropics). With SAoS, the variability is distributed across a range of timescales, from 2 days to 90 days. However, with finer resolution (3-hourly) observations, it is seen that (besides the seasonal cycle) both methods agree and yield a dominant diurnal scale. Along with other subseasonal scales, the contribution and geographical distribution of diurnal scale variability is estimated and shown to be highly significant. Given its large contribution to the variability of tropical rainfall, the diurnal cycle is extracted by means of a Fourier-based filtering and analysed. The diurnal rainfall anomaly is constructed by eliminating all timescales larger than 1 day. Following this, taking care to avoid spurious peaks associated with Gibbs oscillations, the time of day (called the peak octet) when the diurnal anomaly is largest is identified. The peak octet is estimated for each location in the global tropics. This is repeated for 13 years, and the resulting mode of the time of maximum rainfall is established. It is seen that (i) most land regions receive rainfall during the late afternoon/early evening hours; (ii) rainfall over open oceans lack a dominant diurnal signature with a possible combination of early morning and afternoon showers; (iii) coastal regions show a clear south/southwest propagation in the mode of the peak octet of rainfall. In addition to being a comprehensive documentation of the diurnal cycle at very fine scales, the results serve as a critical test for the validation of theoretical and numerical models of global tropical rainfall.