Quantum systems have started to emerge as a disruptive technology and enabling platforms - exploiting the principles of quantum mechanics - to achieve quantum supremacy in computing. Academic research, industrial projects (e.g., Amazon Braket), and consortiums like 'Quantum Flagship' are striving to develop practically capable and commercially viable quantum computing (QC) systems and technologies. Quantum Computing as a Service (QCaaS) is viewed as a solution attuned to the philosophy of service-orientation that can offer QC resources and platforms, as utility computing, to individuals and organisations who do not own quantum computers. To understand the quantum service development life cycle and pinpoint emerging trends, we used evidence-based software engineering approach to conduct a systematic mapping study (SMS) of research that enables or enhances QCaaS. The SMS process retrieved a total of 55 studies, and based on their qualitative assessment we selected 9 of them to investigate (i) the functional aspects, design models, patterns, programming languages, deployment platforms, and (ii) trends of emerging research on QCaaS. The results indicate three modelling notations and a catalogue of five design patterns to architect QCaaS, whereas Python (native code or frameworks) and Amazon Braket are the predominant solutions to implement and deploy QCaaS solutions. From the quantum software engineering (QSE) perspective, this SMS provides empirically grounded findings that could help derive processes, patterns, and reference architectures to engineer software services for QC.