The enzymatic addition of a sequence of adenylyl residues at the 3' end of a non-coding RNA (ncRNA) molecule. In eukaryotes, substrates include nuclear non-coding RNAs such as precursors and a variety of incorrectly processed forms of snRNAs, snoRNAs, rRNAs, and tRNAs, as well as discarded RNA fragments which have been removed from ncRNA primary transcripts. Polyadenylation of precursors is often linked to termination of transcription, but polyadenylation of RNAs targeted for degradation may also occur post-transcriptionally. This polyadenylation is important both for 3'-end processing to produce mature ncRNA species and also for targeting incorrectly processed or discarded RNA molecules for degradation.