Significance In RNA metabolism, many proteins act on diverse cellular RNAs, but avoid certain RNA subsets. To understand how this is accomplished, we investigate the exonuclease Rrp6p, which acts on many, but not all RNAs. We measure degradation kinetics at single-nucleotide resolution and find that Rrp6p discriminates against RNAs ending in CCA-3′. This sequence occurs at the 3′ termini of uncharged tRNAs, a class of abundant RNAs that Rrp6p does not degrade in the cell. Our data provide an unexpected perspective on substrate specificity of RNA-interacting proteins by showing that specificity is used not only to enable preferential action on certain RNAs, but also to selectively avoid RNAs.