Starfleet insignia

The Starfleet insignia is a number of distinctive, slightly asymmetrical, arrowhead-shaped pennants adopted by Starfleet as its identifying emblems, used on starship hulls, installations and some uniforms worn by Starfleet personnel.

History
The United Earth Starfleet of the 22nd century used a simple forward-facing "pointer" shaped pennant, with a round backdrop, that seemed to be based on symbols used by NASA in the 20th and 21st centuries. The original design for the Starfleet uniform of that era did not include this symbol, unless it was a crewperson assigned to Starfleet Command, who would wear the insignia as their assignment patch on the left shoulder. In a later uniform revision of the 2160s, personnel wore the Starfleet patch on their right shoulder as well as the left shoulder assignment patch. 

In the early 23rd century, a simplified pennant version of this symbol was used on Federation starships and installations. This symbol for Starfleet was not worn on uniforms, which again only displayed an individual crewperson's assignment patch. Some starships, such as the Defiant, had an assignment patch similar to this symbol. 

By the early 2270s, while installations and starships (such as Epsilon IX station) maintained their individual assignment patches, Starfleet Command had adopted a new Starfleet insignia, a symbol previously used as the assignment patch of the USS Enterprise (NCC-1701), and prior to that, the USS Kelvin. Specifically, the command division insignia, with the MACO five-pointed star contained in the arrowhead, was most often used, although a version omitting the internal star gained prominence as well. This arrowhead shape, with similar asymmetry as the previous incarnation, had roots in history as being used in a basic form on the United States armed forces space command ratings badge and unit insignia, and on UESPA exploration vessels, such as the Friendship 1, in the 21st century. This symbol was now pointed up when worn on the the chest or shoulder of uniforms, as well as displayed in signage. The arrowhead was faced forward when used as hull decoration. This symbol eventually phased out all assignment patches by the late 2270s, and remained in use for some Starfleet divisions well into the 24th century. The metal badge version of this symbol became Starfleet's first combadge as well. 

In the 2340s, a more stylized version of the symbol, the familiar arrowhead without any internal symbology within the shape, and mounted on an oval field, was incorporated into uniforms. A metal badge of this emblem, again serving as a combadge, was worn on the uniform chest. This symbol also became common in starship hull pennants. 

By the 2370s, a newer version was created for use as a combadge, with a stylized trapezoidal background. Starship hull pennants of this era began using a simplified version of the arrowhead alone, without a background. Even with the newer insignia in use, Starfleet Command still used the 2270s circular-backed version with inscribed star shape in their signage and parts of flag officer uniforms. 

Alternate reality
In the alternate reality created by Nero's incursion, Starfleet had done away with the assignment patches by the 2250s. Instead the fleet was united behind a gold arrowhead, slightly modified from the assignment patch of the USS Kelvin, with a shallow arrowhead indentation on the inside (as seen on dress uniforms). The insignia used on starships was a silver arrowhead with a hollow division symbol inside the arrowhead, similar to the assignment patches used aboard the prime timeline's USS Enterprise (NCC-1701). 

Parallel and future versions
In the mirror universe, a dagger impaling the Earth was the symbol of the Terran Empire and as such was used as the insignia of the $𝜇$. 

In the Barash-created false-future illusion experienced by William T. Riker in 2367, the symbol used in the Starfleet combadge consisted of the arrowhead, either silver or gold, with four horizontal bars behind indicating the rank of the individual by the number of bars that were gold or silver. This version was also in use in an alternate quantum reality visited by Worf in 2370. 

In several alternate timelines observed by Starfleet personnel, new variations of this symbol exist. In the 2390s and 25th century, Starfleet personnel have been known to wear a hollow, outlined version of the arrowhead with two irregular stylized vertical trapezoids behind it as their badge. 

Further in the future, 29th century Starfleet has a rotated version of the arrowhead shape, with the bottom half filled in with a mirror of the top point of the shape. In 29th century timeship hull decorations, the arrowhead has been shown pointing backwards from the direction of motion for the first time, probably because of the vessels' capability for time travel. 

Background information
A great deal of fan fiction and other non-canon sources speculate that the Starfleet-wide adoption of the Enterprise emblem was probably to honor the ship and her crew for their successful and historic five-year mission. This has not been referenced on screen, however. Considering that one of Star Trek's long-held influences is the Age of Sail, it may be a reference to the British Royal Navy's adoption of – the particular coloration of the ships of Admiral Horatio Nelson – on all of its ships after the Battle of Trafalgar. In a form of fact emulating fiction (and vice-versa), the US badges have a delta arrowhead very similar to the 1960s-designed Enterprise insignia, while the 22nd century Starfleet insignia seems to be a deliberate synthesis between the TOS era pennant and the NASA logo.

Several patches were sold off on the It's A Wrap! sale and auction on eBay.

Insignie Hvězdné flotily Sternenflottenabzeichen