Note: Many of the entries here will eventually become full-length posts. Some are rough and have not been fully researched. If you have any corrections or would like to add anything, please comment.
Halcyon: calm and peaceful.
Hands-on: When it functions as an adjective (e.g., the activity was hands-on), give it a hyphen.
Happy median vs. happy medium: Happy median makes sense logically, but happy medium is the common phrase meaning a course or solution that avoids extremes.
Headwinds: wind blowing against the path of a ship; figuratively, forces working against progress.
Helter-skelter: (1) chaotic or disorganized, (2) a disorderly confusion.
Herein: in here.
Hereunder: under here.
Hers: her’s is a misspelling.
Heyday: a period of greatest power, popularity, vigor, etc.
Highlighted: not highlit.
Hijinks vs. high jinks: The always-plural noun meaning noisy activity involving mischievous pranks is spelled hijinks in North America and high jinks outside North America. Hijinx, high jinx, and high-jinx also appear to varying degrees.
Hinky: arousing suspicion.
Hitchhike: three h‘s.
Hitherto: until this time. It is mostly an archaism.
Hodgepodge: (one word) a mixture of elements from varying sources.
Hoi polloi: the common people. From Greek the many.
Hold forth: to talk, often pompously or with strong purpose, for a long time.
Holistic vs. wholistic: There is no substantive difference between them in current usage. Holistic is more common in all main varieties of English.
Homonym: words that sound or are spelled the same but have different meanings.
Hooves: not hoofs.
Hopefully: Don’t listen to the naysayers. Hopefully is a perfectly good sentence adverb (i.e., an adverb that modifies a sentence, often betraying the writer’s attitude toward what is being expressed) meaning it is to be hoped that. Be aware, however, that many English traditionalists despise this use of the word. Hopefully they will change their tune now that the influential style guide of the Associated Press has endorsed the usage.
(In the) hopper: In industry and agriculture, a hopper is a storage device used to hold materials that are ready to be use. Figuratively, something that is in the hopper is on its way or ready to begin.
Horse power vs. horsepower: usually one word. But in U.K. English, it’s spelled as two words about a third of the time.
Hyper- vs. hypo-: Hyper-: over or excessive. Hypo-: under or insufficient.