[Abridged] We investigate the frequency of the various spectral types as a function both of the clusters properties and of the galaxies characteristics. In this way, using the same classification criteria adopted for higher redshift studies, we can consistently compare the properties of the local cluster population to those of their more distant counterparts. We describe a method we have developed to automatically measure the equivalent width of spectral lines in a robust way even in spectra with a non optimal signal to noise. Like this, we can derive a spectral classification reflecting the stellar content, based on the presence and strength of the [OII] and Hdelta lines. We are able to measure 4381 of the ~6000 originally observed spectra, in the fields of 48 clusters, 2744 of which are spectroscopically confirmed cluster members. The spectral classification is then analyzed as a function of galaxies luminosity, stellar mass, morphology, local density and host clusters global properties, and compared to higher redshift samples (MORPHS and EDisCS). The vast majority of galaxies in the local clusters population are passive objects, being also the most luminous and massive. At a magnitude limit of Mv<-18, galaxies in a post-starburst phase represent only ~11% of the cluster population and this fraction is reduced to ~5% at Mv<-19.5, which compares to the 18% at the same magnitude limit for high-z clusters. Normal star forming galaxies [e( c )] are proportionally more common in local clusters. The relative occurrence of post--starbursts suggests a very similar quenching efficiency in clusters at redshifts in the 0 to ~1 range. Furthermore, more important than the global environment, the local density seems to be the main driver of galaxy evolution in local clusters, at least with respect to their stellar populations content.
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