Originally Posted by Sparkle

Not on a regular basis... but the two times (of note) I've heard it mentioned are:

WWII, certainly my grandparents and their peers referenced the special relationship between the US and the UK; with genuine gratitude and a sense of comradery. Similarly the English I know from the same generation - do as well - though not without noting how long it took the US to get involved in WWII.

My great Uncles laughed uproariously when they first heard the phrase:
"Oversexed, overpaid & over here" in reference to their time as GIs stationed in the UK.

The second time I noticed the phrase "special relationship" had become more common was during George W.'s tenure... and it is used in a very sarcastic/sardonic/ironic manner.

George W.'s wars against terror and WMDs and Tony Blair's willingness to follow (presumably because of that "special relationship" - has been widely criticized all around the world.