The team of Service Design Melbourne lists these journals:
Touchpoint: the magazine of the Service Design Network
Here are other journals that have published papers on Service Design in the past based on a research made with Google Scholar:
Journal of Operations Management, with as an example this paper called "The service concept: the missing link in service design research?"
Journal of Service Research with as an example this paper called "Service Design for Experience-Centric Services"
Swedish Design Research Journal with as an example this paper called "Service Design Research Pioneers. An overview of Service Design research developed in Italy since the ‘90s"
Design Issues with as an example this paper called "The Object of Service Design"
The Design Journal with as an example this paper called "Emerging Issues in Service Design"
Public Management Review with as an example this paper called "Design for experience – a public service design approach in the age of digitalization"
Journal of Business Research with as an example this paper called "Service system transformation through service design: Linking analytical dimensions and service design approaches"
The Service Industries Journal with as an example this paper called "Combining Service Blueprint and FMEA for Service Design"
Nordes Nordic Design Research with as an example this paper called "Existing Prototyping Perspectives: Considerations for service design"
Journal of Service Management with as an example this paper called "Linking service design to value creation and service research"
Behaviour & Information Technology with as an example this paper called "Storytelling Group – a co-design method for service design"