What is the single most important criterion when choosing a microscope?

If you are interested in a more technical answer: Look for two things in microscopes. The microscope should be heavy and the objectives should be 160mm DIN standard (or infinity objectives, if you can afford them). Look at the objective of the microscope and check if it has the number 160 written on it. This refers to a 160 mm tube length. Microscopes that are able to accept these optics often (but not always) have a minimum quality. Most educational and routine microscopes use these, plastic toy microscopes do not. These objectives are generally interchangeable with each other. Microscopes that use infinity corrected objectives have an infinity sign printed on them and are expensive and can be found more on research microscopes. Infinity objectives from different manufacturers are not compatible with eah other. I just mention this for the sake of completion. The material of which the body of the microscope is made is also relevant. Devices made mostly of plastic can be considered toys, and these do not provide the stability and optical quality to keep you interested over a longer time period.