RNA splicing has historically been thought to be highly efficient and accurate, with little opportunity for deviation from regulated alternative splicing. This dogma has been challenged by recent observations that biological noise may contribute substantially to transcriptome diversity. However, quantitative understanding of stochastic splicing variation is challenging because these transcripts are likely subject to rapid degradation. Here, we use deep sequencing across RNA compartments to track splicing intermediates in human cells and see abundant cryptic splicing associated with genomic features that promote splicing noise. We observe pervasive usage of low-fidelity splice sites, likely due to stochasticity in recruitment or binding of the spliceosome. These sites are turned over quickly and show evidence for nuclear and cytoplasmic degradation, suggesting widespread surveillance and rapid quality control of non-productive transcripts. Our findings provide insights into the propensity for error in RNA processing mechanisms and regulation of alternative splice sites across a gene.