Update 13 Jan 2016: JIRA Service Desk now has a public REST API. See comment below.
As outlined in this answer question https://answers.atlassian.com/questions/225223/does-jira-service-desk-have-a-public-rest-api# JIRA Servicve Desk does not currently have a public REST API.
We do have internal REST resources that we use in the agent view and we plan to expand that into the "customer portal view" soon but these do not have the strong API guarantee against change. They are there to support the "dynamic browser front end".
Service Desk is of course built on top of the JIRA platform so issue creation / retrieval REST APIs are available via standard JIRA facilities.
As the JIRA Service Desk architect I am intrigued to learn more about about what type of REST APIs people would like and what use cases you are trying to satisfy and how they are more specific to JIRA Service Desk than the current JIRA public REST api.
Please use this improvement request on JAC (https://jira.atlassian.com/) or send me a direct email on email@example.com to further discuss what you are after.