The variant ax is used more in the US, but both ax and axe are used.
Omelette IS the preferred spelling.
pajamas (American preferred spelling) pyjamas (British preferred spelling)
The usual spelling is judgment. The spelling judgement is the less-used variant.
lacy
likeable
The spelling usages are both "cigarette" (the preferred modern spelling) and "cigaret".
Either gray or grey are correct spellings. Grey with an e is the spelling preferred by the British, where gray with the letter a is the preferred spelling by Americans.
An American spelling is a spelling of an individual work such as is preferred in the United States as opposed to the United Kingdom or elsewhere.
That is the UK spelling (e.g. cancer) - "tumour". The preferred US spelling is "tumor".
The ax-like hand tool is spelled "hatchet".
The correct spelling is favorite (liked, preferred).
LICENSE = US, LICENCE = UK.