Like all things perfect, perfect rhyme is less common than other, less perfect versions. That’s because of the nature of its composition. For two words to rhyme perfectly, they must be absolutely identical from the point in which each word's stressed vowel occurs to the end of the word. In addition, the sound that manifests immediately prior to that vowel must be different. Most of the time, this will be a consonant but not always.
This means that words like still and quill are perfect rhymes because they are both one-syllable words, the first stressed vowel is i, and the sounds that follow are identical. Contort and report are also perfect rhymes, but leave and believe are not because the sound immediately preceding the long e sound is identical and also because they have different numbers of syllables. Perfect rhymes, which are also called exact, full, or true rhymes, are less common in multisyllabic words with the stress on one of the earlier syllables.
Perfect rhymes appeal to the child in every reader. Perhaps this is because they are the bread and butter — rhymes perfectly with said and mutter — of nursery rhymes, or perhaps the love of linguistic balance is somehow programmed in our DNA. Poets, however, must use perfect rhyme judiciously. Poems that are created entirely on perfect rhyme end up sounding artificially singsongy. Most poets working in rhyme are aware of this danger and play off perfection with rhymes that might be imperfect or slant.
Imperfect rhymes allow a little leeway in terms of syllable length, plurals, or other additional end sounds. For example, a slant rhyme might pair despair with, well, pair; these two words miss perfection in two ways. First, the articulation that immediately precedes "air" is the same; with a perfect rhyme, they would have to be different, as in pair and fair. Secondly, this particular rhyme allows a two-syllable word to dance with a monosyllabic one, which is verboten in the world of perfect rhyme.
Imperfect or slant rhymes are also called half, near, off, or sprung rhymes. In some cases, it’s the consonants alone that are identical, such in as the word pair stare and store. There is much discussion in poetry circles regarding whether homophones, or words that sound identical but have different meanings, are in fact perfect rhymes. Flower and flour is an example of a word pair with clearly different meanings that a poet might legitimately rhyme in a poem; however, most experts agree that these are technically not perfect rhymes as the articulation preceding the stressed vowels are identical.