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 (<1d) radio properties of GRBs, and has obtained some of the earliest radio detections (GRB 130427A at 0.36 and GRB 130907A at 0.51d post-burst). As all Swift GRBs visible to AMI are observed, this catalogue provides the first representative sample of GRB radio properties, unbiased by multiwavelength selection criteria. We report the detection of six GRB radio afterglows that were not previously detected by other radio telescopes, increasing the rate of radio detections by 50 per cent over an 18-month period. The AMI catalogue implies a Swift GRB radio detection rate of >~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.