I'm curious as to what AR variants does your law enforcement agency authorize or forbid and why?

In getting mine to authorize several, we looked into a couple of different things including if the manufacturer would authorize our Colt certified armorers to work on their weapons.  If not, then not authorized.

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I have the patrol carry what they can shoot well with. We have a small dept and my concern is they hit what they shoot at and know how to operate and can qualify with what they carry, so far the carry rifles are bushmaster , colt , mini-14 , as well as a couple of bolt actions, all 223cal. WE also have two older officers (read  calmer) carry bolt scoped 308s.
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police are citizens with more responsibility

Currently my agency allows us to carry Colts and FN's (most of our rifles are Mil Surplus).  There are no specific models approved or prohibited, just the manufacturer is specified.  

The only other rifle related rule relates to qualifications.  Our regs specifically say that you must qualify with iron sights.  Optics aren't prohibited but you must qualify with the irons.

Our Dept's policy states that the weapon must be the same in form and function as the Colt AR-15 and only in .223 caliber(this allows for any manufacturer such as Bushmaster, Oly Arms, DPMS etc., etc.).  Unfortunately it does not include such variants as the ZM AR-15's due to the different gas/piston system.  We can have optics and upgraded handguards/grips but no two stage or modified triggers.  All weapons must be inspected and recorded by our department armorer and you must qualify with the weapon as you would carry it.  I think it is somewhat important to keep things uniform in the field so that all officers are familiar with the function of another officer's weapon.  Due to liability issues, our agency also issues all officers with AR-15's duty .223 ammunition which is frangible/tactical and will rarely exit your target (bad guy).
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We dont have a police limiting the types of weapons. It's up to the administration on a case by case basis. We have an old timer who carries a lever action 30-30! No kidding. I'm assigned a bolt gun on SERT/SWAT Team but I also carry my own COLT M4. I'm also issued a Benelli Super 90 Tactical w/ an 11" barrel as my duty shotgun. I have ALOT of "Junk in the Trunk"!!

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DoorKicker6, clear..

My department (L.A. Sheriff) only utilizes three different companies to choose from:

Colt, Colt and Colt.  (I don't understand why either???).

Basically, patrol gets the standard lenth Colt AR-15's, while the narcotics units gets the Colt M4's...and the SWAT team uses the new Colt "Commandos" which I understand have an overall length of about 4" shorter than the M4's.  They say the New Commandos have selector switches and are better suited for CQB conditions as compared to the longer M4's, which is important for entry teams.

I wish I could carry my Bushy or my Armalites.

Any privately owned AR is allowed at my dept. as long as it has iron sight capabilities.  There are Colts, Bushmasters, Armalite/Eagles and a couple of parts guns on the road.  Only one guy has a high speed M4 Flat top with a Reflex on it.  Most are pretty plain-Jane.  There are also about a dozen dept issued FN M16A2s out.

All of patrol is issued a Colt M4, and SWAT gets the Commando with 10.5" barrel.  I carry my privately owned Armalite LEO M4 with Reflex, as I bought it while still employed with my old agency.  Deputies may carry their own pesonal rifles as long as they are chambered for .223, and they qualify with it.

Colt produced AR15's only, in factory configuration but can have night sights, scopes, bipods, flashlights, slings and mags up to 40 rounds.  Can be carbine or rifle but nothing shorter than 16" barrel.  The department has issued nearly all of patrol an AR15A3 Tactical Carbine.  I have my own 6920 Law Enforcement Carbine, also known as the semi auto M4 version.  I also have a Beta C-Mag that I carry, even though it isnt specifically authorized, since the department signed the certifying statement letter for me to get it.  Usually I use 30 rounds Colt mags though.  My carbine has a Tactical Tailor sling, Trijicon night sights and a SureFire barrel mounted flashlight and it is about perfect for me.  I have used it twice on arrests and it has just enough accessories to help without having too many to cause problems.  A flashlight and night sights are a MUST on a patrol carbine in my opinion.