Irregularities in the current-voltage-characteristics of a-Si:H based potential barriers have been investigated experimentally, and are discussed theoretically with respect to different transport mechanisms. The investigated samples were different series of double and single barrier a-Si:H - a-Si1-xCx:H - heterostructures, as well as homogeneous samples without heterostructure barrier. Current-voltage-(I-V)-characteristics showing a wide variety of features, from complete smoothness of the curves, to bumps and even accidental step-like switching behaviour, as well as different forms of noise, were recorded at different temperatures. Resonant tunnelling as an explaining transport mechanism for the anomalies was excluded because of inconsistency between experiment and calculations partially including special amorphous features. Instead it is argued that all observed irregularities, i. e. bumps in I-V-curves, switching-like behaviour, and appearance of noise, are related to current transport via trap-assisted tunnelling through locally strongly confined transport paths, leading to the meta-stable formation, change and break-down of conductory filaments.