We present near-IR spectroscopy of a sample of 30 IRAS sources recently identified as late AGB stars, post-AGB stars or early PNe. The spectra obtained are centered at various wavelengths covering the molecular hydrogen v=1-> 0 S(1) 2.122 mu m and v=2->1 S(1) 2.248 mu m emission lines, the recombination lines of hydrogen Brgamma 2.166 mu m, Pfgamma 3.741 mu m and Bralpha 4.052 mu m, and the CO[v=2->0] first overtone bandhead at 2.294 mu m. As a result of these observations we have detected molecular hydrogen emission for the first time in 9 of these sources and confirmed a previous detection by Weintraub et al. (cite{Weintraub98}). This increases from 4 to 13 the total number of proto-PNe detected in molecular hydrogen. In most cases, the positive detections also show emission in the recombination lines of hydrogen (with the exception of IRAS 17150-3224) indicating that the onset of molecular hydrogen emission takes place in the post-AGB phase, very shortly before the nebula becomes ionized. When the molecular hydrogen is fluorescence-excited the detection rate is found to be directly correlated with the evolutionary stage of the central star, rather than with the nebular morphology. When the temperature of the central star is hot enough, fluorescence excitation can be induced by the absorption of UV photons escaping from the rapidly evolving central post-AGB star. In contrast, shock-excited molecular hydrogen is detected only in strongly bipolar proto-PNe, sometimes even at an early stage in the post-AGB phase. Shock excitation is the consequence of the interaction of the fast post-AGB wind with the slow wind material ejected during the AGB. The strong correlation of shock-excited molecular hydrogen emission with bipolarity found confirms the result previously reported by Kastner et al. (cite{Kastner96}) in evolved PNe. However, our results show that this correlation does not exist in the case of fluorescence-excited molecular hydrogen. Based on observations collected at the European Southern Observatory (La Silla, Chile).