Read the riddle once to yourself; then read it again out loud. Make a list of all the obvious answers. As you determine which are most likely incorrect, cross them out.
Examine the riddle for double meanings. For example, in the riddle, "It sits above the crown, though queens never wear it," the words "it sits" could refer to a person, hierarchy, something metaphorical, or a garment or piece of headwear.
Re-read the riddle and consider all immediate and far-fetched possibilities, outside of people, places and things. For example, in the riddle, "I am the beginning of the end, and the end of time and space. I am essential to creation, and I surround every place. What am I?", the correct answer is the letter "e."
Look for hidden clues by re-reading the riddle. Most obvious clues in a riddle seek to fool the reader, but hidden clues are subtleties of language and word choice that help point to the answer. For example, consider the clues in the riddle, "What always runs but never walks, often murmurs, never talks, has a bed but never sleeps, has a mouth but never eats?" A big clue is that it has a bed but never sleeps. You need to start considering objects apart from humans and animals that have beds, like flowers and rivers (and rivers is the correct answer.)