Sunspot number, sunspot area and sunspot unit area are usually used to show sunspot activity. In this paper, periodicity of sunspot activity of modern solar cycles has been investigated through analyzing the monthly mean values of the three indices in the time interval of May 1874 to May 2004 by use of the wavelet transform. Their global power spectra and local power spectra are given while the statistical tests of these spectra are taken into account. The main results are (1) the local wavelet power spectrum of the sunspot number seems like that of the sunspot area, indicating that the periodicity of the both indices is similar. The local power spectrum of the sunspot unit area resembles the local power spectra of the previous two indices, but looks more complicated. (2) the possible periods in sunspot activity are about 10.6 (or 10.9 years for the sunspot unit area), 31, and 42 years, and the period of about 10.6 years is statistically significant in the considered time. For the periods of about 31 and 42 years, their power peaks are under the 95% confidence level line but over the mean red-noise spectral line, and for the other rest periods, their power peaks are even under the mean red-noise spectral line, which are statistically insignificant. (3) the local power of the three periods is higher in the late stage than in the early stage of the considered time. (4) the period characteristics of the three indices, shown in the global power spectra and the local power spectra, are similar but there is difference in detail.