I won't go into the English versus American English spelling debate here, but point out a few editing errors that are very common to both. Ones the computer's spellchecker is unlikely to pick up on. This isn't an extensive list, just the ones I know have come up when I've been editing. There are probably a lot more.

Alright = it doesn't really exist, use all right instead.

Key = a tool to unlock something
Quay = a harbour

Ear = organ of hearing
Ere = before

It's = a shortened form of it is.
Its = belonging to it.

There = a place
Their = belonging to them
They're = shortened form of they are

Here = a place
Hear = the act of hearing something

Hair = the stuff on your head
Hare = an animal similar to a rabbit

Pale = wan in colour
Pail = a bucket

Your = belonging to you
You're = shortened form of you are

You = a person
Yew = a tree
Yore = past

Were = past tense of the verb is
We're = shortened form of we are
Wear = what you do with clothes
Ware = goods for sale
Weir = a small dam across a river or canal

His = belonging to him
Hiss = noise of disapproval

Hers = belonging to her
Hearse = funeral vehicle

Horse = animal
Hoarse = difficulty in speaking

Write = the act of writing
Right = correct

Time = measurement
Thyme = a type of herb

Bough = branch of a tree
Bow = a ribbon or curtsey

Rough = not gentle
Ruff = part of a collar

Steal = take something without permission
Steel = a metal

Stile = steps to help climb over a fence
Style = clothing and accessories

Die = cease living, singular of dice
Dye = change the colour of something

Road = a path to get somewhere
Rode = past tense of ride

Shoe = something for the feet
Shoo = chase something away
Sioux = a native American tribe
Sue = take to court or a girl's name

Too = as well as, also
To = used with a verb, e.g. to ride, to walk
Two = a number

Tow = drag something
Toe = on the feet

Feet = at the bottom of legs, a measurement
Feat = an act of skill or daring

Off = e.g. to turn the cooker off
Of = used such as 'he was an actor of great skill'.

One = a number
Won = to win something

Tents = plural of tent
Tense = upset or nervous

Place = a location
Plaice = a fish

Sore = hurting
Soar = to fly high

Oar = paddle for a boat
Ore = raw metal

Bore = not interesting or drill a hole
Boar = a wild pig

Roux = a base for a sauce
Rue = to regret

Naval = to do with the navy
Navel = the bellybutton

Read = look at a book
Reed = a type of plant

Him = a male
Hymn = a song of praise/worship

Sun = orb in the sky
Son = male child

Team = a group of people
Teem = abundant, e.g. it's teeming with rain

Have = own something
Halve = divide something in two.

Wine = an alcoholic drink
Whine = to complain

Peak = top of something
Peek = to look

Sew = use a needle and thread
So = as in 'so what?'