If I remember correctly the poly-A tails of mRNA are there to protect the ends from nuclease digestion and to serve as a signal for proteins that target it for nuclear export towards the ribosomes for translation. In rRNA/tRNA it would only be present if the protein has several Lysine residues because "AAA" is the codon for Lysine. If I'm not mistaken Lysine residues are usually targeted for acetylation and phosphorylation so it will most likely be part of proteins that are regulated through one of those processes, like in histones.
The 'PolyA' tails are not encoded in the gene and added later by PolyA Polymerase. The genes that contain a PolyA signal and a PolyA site generates transcripts which can be polyadenylated, however there may be additional factors involved in the process. It may be possible to insert such sequences in genes lacking natural PolyA tails. Unlike mentioned by @Loius above, the rRNA and tRNA are noncoding RNAs ane do not code for any protein. The presence of 'AAA' codon and relation to Poly A tail is not clear.