The idiom at loggerheads, which usually functions as a predicate adjective, means in a dispute. Its origins are mysterious. Loggerhead originally referred to a stupid person, and in the 17th century it took a new definition—thick-headed iron tool. When at loggerheads came about soon thereafter, it may have referred to the use of loggerheads as weapons in fights. In any case, at loggerheads (loggerheads is always plural in the idiom) now implies harsh disagreement but not necessarily violence.
The idiom is likely British in origin, and it is used most frequently in British English, but it does appear in all major varieties of English.
The singular loggerhead is mostly archaic. It now appears only in reference to the loggerhead turtle, a species of large sea turtle with a reddish-brown shell.
The two sides remain at loggerheads, and Orbitz hasn’t included any business from American. [Wall Street Journal]
London Underground (LU) and an industrial union are at loggerheads over six planned Tube strikes. [BBC News]
How can a government divided between two parties at loggerheads possibly take the painful decisions needed to tame America’s deficit? [The Economist]
This week the council found itself at loggerheads with the Prime Minister over its criticism on the proposed carbon tax. [Sydney Morning Herald]