What are good free online dictionary websites that have example sentences?
Some of the most reliable online dictionaries are Oxford Advanced Learner's Dictionary, Cambridge Advanced Learner's Dictionary, and Longman's Dictionary of Contemporary English. The best catalogue type dictionaries that list definitions from established Dictionaries with which they have contractual agreements--like American Heritage Dictionary, Random House Dictionary, and Collins English Dictionary--are Dictionary.com and TheFreeDictionary.com.
Using "happy" as an example, Oxford Advanced Learner's Dictionary offers many sentences and separates them visually by major definition, e.g., happy: satisfied. In addition, the definitions follow the somebody/something style: e.g., "willing: happy to do something."
Cambridge Advanced Learner's Dictionary offers sample sentences and offers additional definitions of "happy" and "happy"-related words by word class, e.g., "happy adjective (pleased)."
Longman's Dictionary of Contemporary English offers detailed definitions (e.g., "having feelings of pleasure, for example because something good has happened to you or you are very satisfied with your life") and syntax helps. They offer many sample sentences in addition to providing the somebody/something style: e.g., "be/feel happy for somebody."