I was wondering what type of military personnel is allowed (required) to wear civilian attire for their daily duty. The ones that come to mind are members of the military law enforcement community and intelligence personnel (in both cases, of course, depending on the specific job/assignment). Anybody else?
Elite special forces like the S.A.S. who have a covert mission as their main role.
U.S. forces tend not to since they like to act as large units, not four man undercover teams.
This is not meant to be snarky, but it’s hard to know, since they will look just like the rest of us. The purpose of wearing a uniform is to be recognisable. Many military roles these days probably require anonymity.
(bolding mine)
Hence the question … 
I did for a while. I was a desk clerk for a facility on post which housed personnel and their families until permanent housing could be arranged. To simulate a hotel (I’m still not sure why), we wore sport coats and ties. That was 40 years ago, and I haven’t worn a tie since.
Those are not part of daily duty, but the regulations are interesting.
I was in the Navy and serving at a Naval Air Station back in the mid 70’s. Our shop was half military, half civilian. We were allowed to wear civilian clothes to work on Fridays unless we had duty that day.
David Brinkley wrote in Washington Goes to War (good book about how the capital dealt with WWII) that military personnel were often ordered, in the years before Pearl Harbor, to come and go in Washington in their civvies. FDR, who advocated preparedness but had to deal with rampant isolationism, was worried that there might be a backlash if Congress and the public perceived a marked increase in uniformed personnel in the city. Eisenhower also wrote about being called to a meeting with Gen. George C. Marshall and told to wear a suit, not his uniform, IIRC.
I never knew that military personnel were allowed civilian employment or that they were “off-duty” to the extent it was possible.
It’s allowed with commander’s permission, and with the understanding that your civilian job is secondary to your military one. Five guys in a platoon I was in were fired from their civilian jobs because they had to leave in the middle of the shift because some joker let off a fire extinguisher in the barracks one evening.
Uncle Sam trumps Walmart (or wherever).
Civilian employment is allowed at the discretion of the member’s commander. Not everyone asks for permission, though.
To the OP, there are certain training environments where civilian attire is required; if they’re sending you to a local college course, for instance, you’re on duty but you’re usually expected to dress like the locals. Or if you’re attending a class run by a non-DoD agency, there’s a good chance you’ll be asked to wear whatever the standard of dress is for that agency (say, a suit).
There are also uniforms that don’t really look like uniforms. The guy checking you in at the base hotel might be wearing an official uniform that consists of a polo shirt, slacks, and a nametag.
CID agents in the Army often wear plainclothes during their investigations.
I have never been in the military, but I have been around people who have. I have absolutely never heard of military police wearing anything other than military uniforms.
In my OP I wrote “military law enforcement community” which would include organizations like the Navy NCIS or the Army CID.
On the other hand, I know that for instance in Germany, “regular” MPs (there is actually no equivalent for the Army CID in Germany) would routinely put on civilian clothes when they were looking for soldiers (usually conscripts) who had gone AWOL.
AF Office of Special Investigations (AFOSI) personnel (similar to Army CID) wear civilian clothes. Some others wear civilian clothes depending on their assignment. When required to wear civilian clothes, a civilian clothes allowance is given which is pretty cool. Certain White House duties, or other similar type positions require civilian clothes to be worn on duty.
Sometimes civilian clothes are worn back and forth to work, and then are changed into uniforms once at work. I did this when I was deployed to extreme hardship duty in Budapest during the Kosovo war 
From person experience in the Navy: inspectors/auditors from Naval Reactors (NAVSEA08), when going to a nuclear sub/carrier for an inspection or audit, wear civilian clothes – this is so the inspector/auditor has no visible rank. They might be enlisted, or an officer, or even a civilian, but those being inspected won’t know, and thus won’t treat them any differently.
Another example is the White House Communications Agency (WHCA). When I applied <mumble mumble>* years ago, the inbrief stated that appropriate non-casual civilian business attire was the uniform of the day when working within the White House office spaces. A civilian clothing allowance was part of the package for enlisted members.
We’re talking suits and ties here.
*decades ago. And I didn’t make the cut. But I’m not bitter or anything. 
ETA: I suspect the principle is applicable to all elements of the White House Military Office, in which the DoD provides support services to the Administration in a civilian environment.
Military attaché personnel at embassies across the world, Counter-intelligence agents, and CID are the big three that come to mind for the US Army.
Tons of soldiers delivering pizzas, waiting tables, DJing and bouncing on their nights and weekends.