Assuming that we’re not talking about LotR, are any of the counties commonly referred to as shires? Does it matter if we’re talking about a county with -shire as the suffix?
There are quite a few English counties with the suffix -shire. So if you use the county name, shire is part of that; for example Yorkshire and Lancashire.
There are also several counties with the suffix -sex (no, not that meaning!) e.g. Middlesex and Essex.
I suppose the main use of county names is in cricket, where the National League consists of County teams.
From my point of view, ‘Shire’ means the Lord of the Rings. 
Would anyone ever refer to “the shire of Kent” etc?
It would be an odd way of referring to it.
Yeah, not normally. Someone might do it if they were trying to jokingly mimic old fashioned language i.e. “Yeoman of the bar, render me the King’s Pint of foaming ale from the shire of Kent”
(I’ve never heard anyone actually do that, but it’s the only kind of context I can think it would arise)
Not in England. Maybe a couple of towns in Western Australia, but Kent never was a shire.
Not in the least.
Middlesex is no longer a county (abolished in the 1960s), although there is still a Middlesex University and a Middlesex cricket team.
In New South Wales and some other Australian states, local government was divided into council areas - generally urban / town based - and shires - more rural and urban fringe.
That terminology is changing, but it remains in some specific usages. Our beloved Prime Minister Scott Morrison’s seat is centred on Sutherland Shire on Sydney’s edge, which is universally known as ‘The Shire’.
One odd fossilization of “shire”: “sheriff” comes from “shire reeve”, a royally-appointed officer representing the Crown in a shire.
+1
The largest local government area (LGA) is East Pilbara in north east WA
Covering 372,571 km²; East Pilbara is larger than the state of Victoria ( 227,444 km²) or the United Kingdom (242,495 km²) and is home to a bit over 10,000 people, about half of whom live in the mining town of Newman.
Shire is derived from the Anglo-Saxon word for what (Norman) French calls a county. And as in so much else to do with government the Norman usage wins out, when it comes to the legal entity of local government.
So while there’ll be some residual references to “shire”, for local government it’s "county’ even when “shire” is in the county’s name.
Just about the only reference to “shire” in a generic or abstract sense that I can think of is the occasional (but now more or less gone) term “knights of the shires” as a jokey reference to long-serving Tory back-bench MPs, usually from very safe rural constituencies, and considered to be the centre of gravity in the party and (usually) dependably loyal to the leadership (until suddenly they’re not, and there’s a delegation sent for a “quiet word” to the powers that be).
Well that’s too bad since shire is about 100% cooler sounding than county.
I think they use the name for a unit of local government in Australia.
There’s a town of Newman in County Stanislaushire, California.
There is a massively huge horse breed called a Shire.
As I understand it, there are also some counties where the -shire suffix in the name is optional. Devon, for instance, can be called just Devon but also Devonshire (even though the latter seems to be rather archaic).
Sometimes “The Shires” is used as a collective noun for counties, particularly rural counties. It is usually politician or journalist speak.
I was thinking this. The only time I could imagine it being referenced would be someone (jokingly) referring to the countryside. eg ‘Haven’t seen Joe in years. He moved out to the Shires years ago’. Kind of like another word for the Sticks.
But in general answer to the OP, no, we don’t drop ‘shire’ into everyday language.
There are still a few municipal buildings around called Shire Hall. I know the one in Reading, Berkshire.
TCMF-2L
Wikipedia says in the US that the states of Vermont, Rhode Island, Massachusetts, and Maine still use the term “shire town” when referring to what is called a “county seat” in other states.
The Society for Creative Anachronism (a medieval re-enactment group) uses the term shire for “a local branch that reports through a principality or kingdom. It is a lot like a canton, except it is not part of a barony.”
There are a few -shires within Yorkshire that are not counted among the list of official counties, such as Allertonshire, Richmondshire and Howdenshire. Perhaps the most famous of these is Hallamshire, which gives its name to the local hospital and golf club in Sheffield.
The term ‘shire’ is rarely used by itself, even in these contexts.