Hereditary spastic paraplegia (HSP) is an inherited neurological condition that leads to progressive spasticity and gait abnormalities. Adult-onset HSP is most commonly caused by mutations in SPAST, which encodes spastin a microtubule severing protein. In olfactory stem cell lines derived from patients carrying different SPAST mutations, we investigated microtubule-dependent peroxisome movement with time-lapse imaging and automated image analysis. The average speed of peroxisomes in patient-cells was slower, with fewer fast moving peroxisomes than in cells from healthy controls. This was not because of impairment of peroxisome-microtubule interactions because the time-dependent saltatory dynamics of movement of individual peroxisomes was unaffected in patient-cells. Our observations indicate that average peroxisome speeds are less in patient-cells because of the lower probability of individual peroxisome interactions with the reduced numbers of stable microtubules: peroxisome speeds in patient cells are restored by epothilone D, a tubulin-binding drug that increases the number of stable microtubules to control levels. Patient-cells were under increased oxidative stress and were more sensitive than control-cells to hydrogen peroxide, which is primarily metabolised by peroxisomal catalase. Epothilone D also ameliorated patient-cell sensitivity to hydrogen-peroxide. Our findings suggest a mechanism for neurodegeneration whereby SPAST mutations indirectly lead to impaired peroxisome transport and oxidative stress.