The most comprehensive work on qualification patches is the Cloth Qualification Patch Catalog published by the American Society of Military Insignia Collectors. Steve Tatum was the senior editor, William K. Emerson and I assisted him. The catalog was published in THE TRADING POST, ASMIC's quarterly journal, as follows:
In addition, I authored a detailed two-part article on qualification patches:
"U.S. Army Tank Gunnery Qualification Patches." THE TRADING POST. Part 1, April-June 2009, pp. 39-44 Part 2, July-September 2009, pp. 37-39
Additional articles published in THE TRADING POST that addressed qualification patches:
"3rd Armored Division Qualification Insignia," THE TRADING POST (Oct-Dec 1973), pp. 71-72.
SFC John Huffman, "Qualification Insignia," THE TRADING POST (Jan-Mar 1974), p. 69.
"Qualification Badges," THE TRADING POST (Jan-Mar 1980), pp. 31-32.
William K. Emerson, "Tid Bits," THE TRADING POST (Jul-Sep 1998), pp. 7-8.
Frank Hummel, "1-4 Cavalry Badges," THE TRADING POST (Jan-Mar 2001), p. 14.
Tom Ayers, "Tank Table VIII Qualification Badge," THE TRADING POST (Jul-Sep 2017), p. 14.
Terrance Worman, "U.S. Army 2/68th Armor Ground Surveillance Radar Patch," THE TRADING POST (Jul-Sep 2018), p. 50.
William K. Emerson, "Tid-Bits," THE TRADING POST (Jul-Sep 2020), pp. 3-8.
Various TCQC patches are illustrated in THE TRADING POST as follows:
I authored an article in ARMOR, an official U.S. Army periodical:
"U.S. Army Tank Gunnery Qualification Patches," ARMOR (Spring 2021), pp. 66-69.
3rd Armored Division Regulation No. 672-5, "Decorations, Awards And Recognition" (24 Apr 78)
HQ, III Corps and Fort Hood, Memo re Distinctive Tank Gunnery Patch (25 Nov 80)
HQ, III Corps and Fort Hood, Fort Hood Supplement 1 to Army Regulation 672-5-1, Decorations, Awards, and Honors: Military Awards (29 Jan 87)
HQ, 2nd Armored Division, Memo re 2d Armored Division Infantry Gunnery Circular (Draft) (undated)
HQ, 3rd Armored Division, Memo re Authorized Patches for 3d Armored Division (30 Mar 81)
HQ, 3rd Armored Division, Memo re Division Awards to Distinguished Platoons (20 Apr 81)
HQ, 3rd Armored Division, The Spearhead Standard (1 Mar 83)
HQ, 9th Infantry Division, Soldier's Marksmanship Recognition, Musketry Expert Patch (25 Jun 81)
Fact Sheet, "Tank Gunnery Patches" (Germany: 1st Armd Div Museum, n.d.)
SP4 John Stark, ed., 3rd Armored Division Germany 1963. Marceline, MO: Walsworth, n.d.
1LT S.J. Buchanan, OIC, Spearhead '66. Marceline, MO: Walsworth, n.d.
3d Infantry Division Schweinfurt 1972. Marceline, MO: Walsworth Publishing Co., n.d.
1LT Jacqueline A. Lockhart, Editor-in-Chief, 49th Lone Star Armored Division 1985-86. N.p.: Taylor Publishing Co., n.d.
4th Squadron 7th US Cavalry Camp McKenzie 1968 Korea. Seoul, ROK: Hyun Dae Commercial Co. Ltd., 1968.
1st Bn, 72d Armor Camp Casey Korea 1973. N.p.: n.p., 1973(?).
Crusaders Forged In Fire Korea 1974 Second Infantry Division. N.p.: n.p., 1974(?). (1-72 Armor).
Crusaders Do It Better Forged In Fire Spring Korea 1975 Second Infantry Division. N.p.: n.p., 1975(?). (1-72 Armor).
Crusaders Do It Better Forged In Fire Spring 1978 Korea Second Infantry Division. N.p.: n.p., 1978(?). (1-72 Armor).
First Tank Second Infantry Division Camp Casey Korea 1983. N.p.: n.p., 1983(?). (1-72 Armor).
First Tank Second Infantry Division Camp Casey Korea 1984. N.p.: n.p., 1984(?). (1-72 Armor).
2ND Battalion, 72D Armor 2D Infantry Division Camp Beard 1969-1970 Korea. N.p.: Hyon Dae Commercial Co. Ltd., 1970(?).
1st Bn, 73rd Armor Camp Casey Korea – 1971. N.p.: n.p., 1971(?).
1st Battalion 73d Armor Camp Casey Tong Du Chun, Korea-1972. N.p.: n.p., 1972.
1st Battalion 73d Armor Thunderbolts Camp Casey-Korea-1974. N.p.: n.p., 1973.
Letter to the Author from HQ, 2-77 Armor (17 Jun 77).
Correspondence from various other Army units
U.S. Militaria Forum: www.usmilitariforum.com
The main thread is titled: "TCQC patches"
There are some other threads; search for "TCQC" to find them
A website commemorating the 1st Bn, 22nd Infantry includes illustrations of some qualification patches worn in the 4th Infantry Division
Click on "Insignia & Memorabilia," then "4th Infantry Division Insignia Part 4"
There are a fair number of websites that include a random photograph or two of a qualification patch, or of soldiers wearing qualification patches, typically without any explanation.
If you search for "TCQC" or "Army Qualification Patches" you will find a fair number of illustrations of gunnery patches, but without any identification or explanation of the patches.
The webpage for the USA Brotherhood Of Tankers (USABOT) used to illustrate a large number of qualification patches (circa 2016), but no longer includes any of those illustrations. www.usabot.org
Surprisingly, there do not appear to be any decent sources on the Internet that discuss gunnery patches in any sort of comprehensive manner.