Significance Glioblastoma is the most common type of adult brain cancer, with near-uniform fatality within 2 y of diagnosis. Therapeutic failure is thought to be related to small subpopulations of cells that exhibit tumorigenicity, the cellular capacity to reconstitute the entire tumor mass. One fundamental issue is whether tumorigenicity exists within a static subpopulation of cells or whether the capacity is stochastically acquired. We provide evidence that tumorigenicity is a cellular property that is durable yet undergoes low-frequency stochastic changes. We showed that these changes are driven by lysine-specific demethylase 1 (LSD1)-mediated epigenetic (heritable non-DNA sequence-altering) modifications that impact expression of key transcription factors, which in turn govern transitions between tumorigenic states. These findings harbor implications for glioblastoma therapeutic development.