Amyloid fibril formation is central to the pathology of many diseases, including neurodegenerative disorders such as Alzheimer's and Parkinson's disease. Amyloid fibrils can also have functional and scaffolding roles, for example in bacterial biofilms, and have also been exploited as useful biomaterials. Despite being linear protein homopolymers, amyloid fibrils can exhibit significant structural and morphological polymorphism, making it relevant to study them on the single fibril level. We here introduce the concept of nanofluidic channel analysis to the study of single, fluorescently-labeled amyloid fibrils in solution, monitoring the extension and emission intensity of individual fibrils confined in nanochannels with a depth of 300 nm and a width that gradually increases from 300 to 3000 nm. The change in fibril extension with channel width permitted accurate determination of the persistence length of individual fibrils using Odijk's theory for strongly confined polymers. The technique was applied to amyloid fibrils prepared from the Alzheimer's related peptide amyloid-β(1-42) and the Parkinson's related protein α-synuclein, obtaining mean persistence lengths of 5.9 ± 4.5 μm and 3.0 ± 1.6 μm, respectively. The broad distributions of fibril persistence lengths indicate that amyloid fibril polymorphism can manifest in their physical properties. Interestingly, the α-synuclein fibrils had lower persistence lengths than the amyloid-β(1-42) fibrils, despite being thicker. Furthermore, there was no obvious within-sample correlation between the fluorescence emission intensity per unit length of the labelled fibrils and their persistence lengths, suggesting that stiffness may not be proportional to thickness. We foresee that the nanofluidics methodology established here will be a useful tool to study amyloid fibrils on the single fibril level to gain information on heterogeneity in their physical properties and interactions.