We present the Arcminute Microkelvin Imager (AMI) Large Array catalogue of 139 gamma-ray bursts (GRBs). AMI observes at a central frequency of 15.7GHz and is equipped with a fully automated rapid-response mode, which enables the telescope to respond to high-energy transients detected by Swift. On receiving a transient alert, AMI can be on-target within 2-min, scheduling later start times if the source is below the horizon. Further AMI observations are manually scheduled for several days following the trigger. The AMI GRB programme probes the early-time (~15 per cent, down to ~0.2mJy/beam. However, scaling this by the fraction of GRBs AMI would have detected in the Chandra & Frail (2012, Cat. J/ApJ/746/156) sample (all radio-observed GRBs between 1997 and 2011), it is possible ~44-56 per cent of Swift GRBs are radio bright, down to ~0.1-0.15mJy/beam. This increase from the Chandra & Frail (2012, Cat. J/ApJ/746/156) rate (~30 per cent) is likely due to the AMI rapid-response mode, which allows observations to begin while the reverse-shock is contributing to the radio afterglow.