The paper explores the essence of the criteria and indicators which can be used to select a cloud-oriented learning support system for a higher education institution. The following criteria with corresponding indicators are identified: design criterion (reliability, accessibility, multilinguality, security, adaptability, ease of use and administration, free use); technological criterion (user access rights differentiation, cloud storage of data, integration with other cloud-based services, ability to download different types of files); communication criterion (user registration, communication between registered users, creating groups, creating forums and chats); information-didactic criterion (structuredness, calendar, assessment of student achievement, file sharing, testing and surveys, group and individual modes of work; analytics for a particular course). The most downloaded LMS are shown based on the results published by LMS Market Share. The paper offers an analysis of a number of cloud-based learning management systems (Google Classroom, Moodle, Edmodo, Studyboard, Oracle, Learner Nation, iSpring, Canvas, Schoology, Blackboard, NeoLms) in terms of the above-mentioned criteria and indicators. The systems were selected based on the method of expert evaluation. The expert evaluation showed that the most convenient and high-quality cloud-based learning management system for building a cloud-oriented learning environment of a higher education institution which best meets all the criteria are NeoLMS, Canvas and Google Classroom. These LMS offer all the functionalities which are essential in the educational process. We see the development of methodological recommendations for higher education regarding the high-quality and successful implementation of such learning management systems in the educational process as prospects for further research.