Radio signal recognition is a crucial task in both civilian and military applications, as accurate and timely identification of unknown signals is an essential part of spectrum management and electronic warfare. The majority of research in this field has focused on applying deep learning for modulation classification, leaving the task of signal characterisation as an understudied area. This paper addresses this gap by presenting an approach for tackling radar signal classification and characterisation as a multi-task learning (MTL) problem. We propose the IQ Signal Transformer (IQST) among several reference architectures that allow for simultaneous optimisation of multiple regression and classification tasks. We demonstrate the performance of our proposed MTL model on a synthetic radar dataset, while also providing a first-of-its-kind benchmark for radar signal characterisation.