DAPI stains A and T nucleotide rich regions, does it mean it stains only DNA not RNA? if it does stain RNA, isn't it possible that it stains microparticles/ microvesicles/ exosomes even if they don't have nuclei?
To partly answer your question, DAPI also binds RNA. However, the DAPI/RNA complex exhibits a longer-wavelength fluorescence emission maximum than the DAPI/dsDNA complex (~500 nm versus ~460 nm) and a quantum yield that is only about 20% as high.
I am not qualified enough for this but you might be interested in looking at this paper which gives further detail http://www.ncbi.nlm.nih.gov/pubmed/1372825
In theory, DAPI should stain microvesicles and exosomes through RNA binding as Norbert Weiss suggests above, however in practice I doubt the labelled RNA fluorescence is strong enough to detect, again for the reasons Norbert mentions.
Notably, some studies show RNA in the larger microvesicles and some show none. Exosomes, contain a relatively high RNA content for their size, but I still doubt the fluorescent signal would be strong enough to detect. I've never seen any significant cytoplasmic staining with DAPI in confocal studies, even when I've really overloaded the DAPI.
I have to admit however, having read a fair bit of extracellular vesicle literature i've never seen DAPI used for this purpose. Instead, mostly membrane-permeable cytoplasmic or lipd-soluble membrane probes are used. Increasingly commonly, fluorescent protein expression vectors are introduced to the secreting cell, which are taken up into the secreted vesicle, allowing them to be tracked.