On the left lapel of the wearer.
The US Army is presently going to dress blues.
There are a bunch of them. Branch of Service insignia, rank insignia and other insignia that aren't unit-specific are found in Army Regulation 670-1, which is downloadable from a few places. Distinguished Unit Insignia, which everyone calls unit crests, and patches are not in that reg; check the Institute of Heraldry website at http://www.tioh.hqda.pentagon.mil/
The length of a dollar bill will be close.
I am guessing around 1900 http://members.tripod.com/~Fbg_mem_museum/uniftalk/unipat2.htm
AR 670-1, para. 29-16a(2) Remove when they "fail to qualify or are exempt from qualification by Army regulations."
Although there are different types of military badges for the US Army, there are no subdivisions for military insignia. The total number of military insignia for the US Army is 57.
AR 670-1 Wear and Appearance of Army Uniforms That will cover everything from greens, acus, whites, blues, how long side burns should be and what gloves go where and when.
Jon A. Maguire has written: 'Silver wings, pinks & greens' -- subject(s): Air forces, History, Insignia, Medals, badges, decorations, Uniforms, United States, United States. Army Air Forces, World War, 1939-1945 'Gooney birds & ferry tales' -- subject(s): American Personal narratives, History, Personal narratives, American, Transportation, United States, United States. Army Air Forces. Air Transport Group, 27th, World War, 1939-1945 'More Silver Wings, Pinks & Greens' -- subject(s): Insignia, Uniforms, United States, United States. Army Air Forces, United States. Army. Air Corps, United States. Army. Air Service, Wings (Insignia)
For the US Army, it's a bronze oak leaf. The same insignia is used by Majors in the Air Force and Marine Corps, and by those holding the rank of Lieutenant Commander in the Navy and Coast Guard while wearing certain uniforms.
A shield with an eagle in the center
The unit crest is worn on the epalette of the dress green uniform,all nco's wear a green cloth secured to the epalette and the unit crest is placed upon the same...
The correct placement of rank for a non-commissioned officer's Army dress blues is centered on the left sleeve between the shoulder seam and the elbow. The shoulder sleeve insignia is centered on the left shoulder 1/2 inch below the top of the shoulder seam.