We present two galaxy shape catalogues from the Dark Energy Survey Year 1 data set, covering 1500 deg(2) with a median redshift of 0.59. The catalogues cover two main fields: Stripe 82, and an area overlapping the South Pole Telescope survey region. We describe our data analysis process and in particular our shape measurement using two independent shear measurement pipelines, METACALIBRATION and IM3SHAPE. The METACALIBRATION catalogue uses a Gaussian model with an innovative internal calibration scheme, and was applied to riz bands, yielding 34.8M objects. The IM3SHAPE catalogue uses amaximum-likelihood bulge/disc model calibrated using simulations, and was applied to r-band data, yielding 21.9M objects. Both catalogues pass a suite of null tests that demonstrate their fitness for use in weak lensing science. We estimate the 1 sigma uncertainties in multiplicative shear calibration to be 0.013 and 0.025 for the METACALIBRATION and IM3SHAPE catalogues, respectively.