Told to avoid the nominative case form "You and me are going. . . ," people will avoid the phrase you and me even when it appears in the oblique case, and will end up saying things like, "Between you and I. . ." Similar confusion surrounds the pronoun whom; people assume that whom is the formal and fancy version, and end up saying things like "Whom shall I say is calling?"
Told that they should never "drop" the ending -ly from adverbs, people produce new words like thusly, soonly, and fastly. Spurious adverb forms also appear behind words that are serving as a copula, and thus would call for a simple predicate in traditional grammar: "my eyes are going badly."
Another area of hypercorrection involves Greek and Latin looking words like octopus; the spurious plural octopi likens the octopus to a number of Latin words that form irregular plurals in -i. (Were there actually a classical plural of octopus, it would be *octopodes.) Platypus, cactus, status, hiatus, rebus, syllabus, mandamus, and caucus are sometimes inflected the same way; none would be inflected that way in Latin or Greek. Virus sometimes gets the even more inappropriate pseudoclassical plural virii, which presumes Latin *virius. All of these words can and should also take the regular English inflection in -es.
When pronunciation of learned words goes astray, it is sometimes called a hyperforeignism. Tell people that the -s is silent in Mardi Gras, and they will imagine that it also falls silent in other learned French phrases, and begin to pronounce coup de grâce as "coo de grah."