EDIT: Please see this post for the most recent July 2021 trophy badge count.
I was curious about which trophies have been awarded the most number of times, so I went through and checked. All badges except custom ones are included, and the numbers are current as of 2021-03-03 4:44 PM GMT:
BTS ‒ BTS: 74
BlackPink ‒ BlackPink: 48
EXO ‒ EXO: 35
Twice ‒ Twice: 23
BlackPink ‒ Rosé: 19
Red Velvet ‒ Red Velvet: 19
BlackPink ‒ Jisoo: 15
BlackPink ‒ Lisa: 15
(G)I-DLE ‒ (G)I-DLE: 15
Girls' Generation ‒ Girls' Generation: 15
Aespa ‒ Aespa: 14
BlackPink ‒ Jennie: 14
BTS ‒ Jungkook: 13
Dreamcatcher ‒ Dreamcatcher: 13
Soloist ‒ IU: 12
Mamamoo ‒ Mamamoo: 11
BTS ‒ Suga: 10
Stray Kids ‒ Stray Kids: 10
Treasure ‒ Treasure: 10
BIGBANG ‒ BIGBANG: 9
BTS ‒ V: 9
TXT ‒ TXT: 9
Twice ‒ Tzuyu: 8
BTS ‒ Jimin: 7
Girls' Generation ‒ Taeyeon: 7
NCT ‒ NCT: 7
Twice ‒ Jihyo: 7
Aespa ‒ Ningning: 6
BTS ‒ RM: 6
Red Velvet ‒ Seulgi: 6
SHINee ‒ SHINee: 6
BIGBANG ‒ G-Dragon: 5
BTS ‒ Jin: 5
EXO ‒ Kai: 5
GOT7 ‒ GOT7: 5
Itzy ‒ Itzy: 5
Seventeen ‒ Seventeen: 5
Twice ‒ Nayeon: 5
Aespa ‒ Winter: 4
Ateez ‒ Ateez: 4
ENHYPEN ‒ ENHYPEN: 4
EXO ‒ Baekhyun: 4
GOT7 ‒ JB: 4
Seventeen ‒ DK: 4
Twice ‒ Mina: 4
Twice ‒ Momo: 4
Ateez ‒ Hongjoong: 3
EXO ‒ D.O: 3
(G)I-DLE ‒ Shuhua: 3
(G)I-DLE ‒ Soyeon: 3
Itzy ‒ Ryujin: 3
Itzy ‒ Yuna: 3
LOONA ‒ LOONA: 3
Mamamoo ‒ Hwasa: 3
MONSTA X ‒ MONSTA X: 3
MONSTA X ‒ I.M: 3
MONSTA X ‒ Wonho: 3
NCT ‒ NCT WayV: 3
NCT ‒ Jungwoo: 3
NCT ‒ Mark: 3
Seventeen ‒ The8: 3
SHINee ‒ Taemin: 3
Stray Kids ‒ Lee Know: 3
Super Junior ‒ Super Junior: 3
Twice ‒ Dahyun: 3
TXT ‒ Soobin: 3
TXT ‒ Yeonjun: 3
Soloist ‒ Heize: 3
Ateez ‒ San: 2
Ateez ‒ Seonghwa: 2
Ateez ‒ Yeosang: 2
BIGBANG ‒ T.O.P: 2
BTS ‒ J-Hope: 2
CLC ‒ CLC: 2
Dreamcatcher ‒ Dami: 2
Dreamcatcher ‒ Gahyeon: 2
Dreamcatcher ‒ Siyeon: 2
Dreamcatcher ‒ Yoohyeon: 2
ENHYPEN ‒ Sunoo: 2
(G)I-DLE ‒ Minnie: 2
(G)I-DLE ‒ Yuqi: 2
Girls' Generation ‒ Tiffany: 2
Girls' Generation ‒ Yoona: 2
GOT7 ‒ Jinyoung: 2
LOONA ‒ Vivi: 2
Miss A ‒ Miss A: 2
NCT ‒ NCT Dream: 2
NCT ‒ Haechan: 2
NCT ‒ Jaehyun: 2
NCT ‒ Jaemin: 2
NCT ‒ Jeno: 2
NCT ‒ Johnny: 2
NCT ‒ Taeyong: 2
NCT ‒ Yangyang: 2
NCT ‒ Yuta: 2
Red Velvet ‒ Irene: 2
Red Velvet ‒ Yeri: 2
Seventeen ‒ Hoshi: 2
Seventeen ‒ Jun: 2
Seventeen ‒ Mingyu: 2
Seventeen ‒ S.Coups: 2
Stray Kids ‒ Felix: 2
Stray Kids ‒ Hyunjin: 2
The Boyz ‒ The Boyz: 2
Treasure ‒ Asahi: 2
Treasure ‒ Choi Hyunsuk: 2
Treasure ‒ Yoshi: 2
Twice ‒ Sana: 2
TXT ‒ Beomgyu: 2
TXT ‒ Huening Kai: 2
TXT ‒ Taehyun: 2
2NE1 ‒ Bom: 1
AB6IX ‒ AB6IX: 1
AB6IX ‒ Daehwi: 1
AB6IX ‒ Donghyun: 1
AB6IX ‒ Woojin: 1
AB6IX ‒ Woong: 1
AB6IX ‒ Youngmin: 1
Ateez ‒ Mingi: 1
Ateez ‒ Wooyoung: 1
Ateez ‒ Yunho: 1
BIGBANG ‒ Daesung: 1
BIGBANG ‒ Taeyang: 1
Day6 ‒ Day6: 1
Day6 ‒ Dowoon: 1
Day6 ‒ Jae: 1
Day6 ‒ Sungjin: 1
Day6 ‒ Wonpil: 1
Day6 ‒ YoungK: 1
Everglow ‒ Everglow: 1
Everglow ‒ Aisha: 1
Everglow ‒ E:U: 1
EXO ‒ Chanyeol: 1
f(x) ‒ Sulli: 1
f(x) ‒ Krystal: 1
GFriend ‒ GFriend: 1
GFriend ‒ Eunha: 1
(G)I-DLE ‒ Miyeon: 1
(G)I-DLE ‒ Soojin: 1
Girls' Generation ‒ Seohyun: 1
Girls' Generation ‒ Yuri: 1
Golden Child ‒ Golden Child: 1
GOT7 ‒ JJ Project: 1
GOT7 ‒ JUS2: 1
GOT7 ‒ Bambam: 1
GOT7 ‒ Jackson: 1
GOT7 ‒ Mark: 1
GOT7 ‒ Youngjae: 1
GOT7 ‒ Yugyeom: 1
GWSN ‒ GWSN: 1
I.O.I ‒ I.O.I: 1
I.O.I ‒ Chungha: 1
Itzy ‒ Chaeryeong: 1
Itzy ‒ Lia: 1
IZ*ONE ‒ Yuri: 1
IZ*ONE ‒ Sakura: 1
LOONA ‒ Chuu: 1
Mamamoo ‒ Solar: 1
Mamamoo ‒ Wheein: 1
Miss A ‒ Suzy: 1
MONSTA X ‒ Kihyun: 1
MONSTA X ‒ Shownu: 1
NCT ‒ NCT 127: 1
NCT ‒ NCT U: 1
NCT ‒ Kun: 1
NCT ‒ Renjun: 1
Oh My Girl ‒ Arin: 1
SATURDAY ‒ SATURDAY: 1
SATURDAY ‒ Yuki: 1
Seventeen ‒ Jeonghan: 1
Seventeen ‒ Seungkwan: 1
Seventeen ‒ Vernon: 1
SHINee ‒ Jonghyun: 1
SHINee ‒ Key: 1
SHINee ‒ Onew: 1
StayC ‒ StayC: 1
StayC ‒ Isa: 1
Stray Kids ‒ Bangchan: 1
Stray Kids ‒ Han: 1
Stray Kids ‒ I.N: 1
Stray Kids ‒ Seungmin: 1
The Boyz ‒ Hyunjae: 1
The Boyz ‒ Younghoon: 1
Treasure ‒ Haruto: 1
Treasure ‒ Junkyu: 1
Treasure ‒ Park Jeongwoo: 1
Twice ‒ Jeongyeon: 1
UP10TION ‒ UP10TION: 1
WINNER ‒ Mino: 1
WJSN ‒ Cheng Xiao: 1
Soloist ‒ BoA: 1
2NE1 ‒ CL: 0
2NE1 ‒ Dara: 0
2NE1 ‒ Minzy: 0
Astro ‒ Astro: 0
Astro ‒ Eunwoo: 0
Astro ‒ Jinjin: 0
Astro ‒ MJ: 0
Astro ‒ Moonbin: 0
Astro ‒ Rocky: 0
Astro ‒ Sanha: 0
Ateez ‒ Jongho: 0
BIGBANG ‒ Seungri: 0
CLC ‒ Elkie: 0
CLC ‒ Eunbin: 0
CLC ‒ Seunghee: 0
CLC ‒ Seungyeon: 0
CLC ‒ Sorn: 0
CLC ‒ Yeeun: 0
CLC ‒ Yujin: 0
Dreamcatcher ‒ Handong: 0
Dreamcatcher ‒ Jiu: 0
Dreamcatcher ‒ SuA: 0
ENHYPEN ‒ Heeseung: 0
ENHYPEN ‒ Jake: 0
ENHYPEN ‒ Jay: 0
ENHYPEN ‒ Jungwon: 0
ENHYPEN ‒ Ni-ki: 0
ENHYPEN ‒ Sunghoon: 0
Everglow ‒ Mia: 0
Everglow ‒ Onda: 0
Everglow ‒ Sihyeon: 0
Everglow ‒ Yiren: 0
EXO ‒ Lay: 0
EXO ‒ Suho: 0
EXO ‒ Chen: 0
EXO ‒ Sehun: 0
EXO ‒ Xiumin: 0
f(x) ‒ f(x): 0
f(x) ‒ Amber: 0
f(x) ‒ Luna: 0
f(x) ‒ Victoria: 0
fromis_9 ‒ fromis_9: 0
fromis_9 ‒ Chaeyoung: 0
fromis_9 ‒ Gyuri: 0
fromis_9 ‒ Jiheon: 0
fromis_9 ‒ Jisun: 0
fromis_9 ‒ Jiwon: 0
fromis_9 ‒ Nagyung: 0
fromis_9 ‒ Saerom: 0
fromis_9 ‒ Seoyeon: 0
GFriend ‒ SinB: 0
GFriend ‒ Sowon: 0
GFriend ‒ Umji: 0
GFriend ‒ Yerin: 0
GFriend ‒ Yuju: 0
Girls' Generation ‒ Hyoyeon: 0
Girls' Generation ‒ Jessica: 0
Girls' Generation ‒ Sooyoung: 0
Girls' Generation ‒ Sunny: 0
Golden Child ‒ Bomin: 0
Golden Child ‒ Daeyeol: 0
Golden Child ‒ Donghyun: 0
Golden Child ‒ Jaehyun: 0
Golden Child ‒ Jangjun: 0
Golden Child ‒ Jibeom: 0
Golden Child ‒ Joochan: 0
Golden Child ‒ Seungmin: 0
Golden Child ‒ Tag: 0
Golden Child ‒ Y: 0
GWSN ‒ Anne: 0
GWSN ‒ Lena: 0
GWSN ‒ Minju: 0
GWSN ‒ Miya: 0
GWSN ‒ Seokyung: 0
GWSN ‒ Seoryoung: 0
GWSN ‒ Soso: 0
iKON ‒ iKON: 0
iKON ‒ B.I: 0
iKON ‒ Bobby: 0
iKON ‒ Chanwoo: 0
iKON ‒ Donghyuk: 0
iKON ‒ Jinhwan: 0
iKON ‒ Junhoe: 0
iKON ‒ Yunhyeong: 0
I.O.I ‒ Sohye: 0
I.O.I ‒ Somi: 0
IZ*ONE ‒ IZ*ONE: 0
IZ*ONE ‒ Chaewon: 0
IZ*ONE ‒ Chaeyeon: 0
IZ*ONE ‒ Hyewon: 0
IZ*ONE ‒ Minju: 0
IZ*ONE ‒ Nako: 0
IZ*ONE ‒ Yena: 0
IZ*ONE ‒ Yujin: 0
IZ*ONE ‒ Hitomi: 0
JO1 ‒ JO1: 0
JO1 ‒ Yonashiro Sho: 0
JO1 ‒ Kawashiri Ren: 0
JO1 ‒ Shiroiwa Ruki: 0
JO1 ‒ Kono Junki: 0
JO1 ‒ Sato Keigo: 0
JO1 ‒ Kawanishi Takumi: 0
JO1 ‒ Kimata Syoya: 0
JO1 ‒ Ohira Shosei: 0
JO1 ‒ Kinjo Sukai: 0
JO1 ‒ Tsurubo Shion: 0
JO1 ‒ Mamehara Issei: 0
LOONA ‒ LOONA yyxy: 0
LOONA ‒ Choerry: 0
LOONA ‒ Gowon: 0
LOONA ‒ Haseul: 0
LOONA ‒ Heejin: 0
LOONA ‒ Hyunjin: 0
LOONA ‒ Jinsoul: 0
LOONA ‒ Kim Lip: 0
LOONA ‒ Olivia Hye: 0
LOONA ‒ Yeojin: 0
LOONA ‒ Yves: 0
Mamamoo ‒ Moonbyul: 0
Miss A ‒ Fei: 0
Miss A ‒ Jia: 0
Miss A ‒ Min: 0
Momoland ‒ Momoland: 0
Momoland ‒ Ahin: 0
Momoland ‒ Daisy: 0
Momoland ‒ Hyebin: 0
Momoland ‒ Jane: 0
Momoland ‒ JooE: 0
Momoland ‒ Nancy: 0
Momoland ‒ Nayun: 0
Momoland ‒ Taeha: 0
Momoland ‒ Yeonwoo: 0
MONSTA X ‒ Hyungwon: 0
MONSTA X ‒ Joohoney: 0
MONSTA X ‒ Minhyuk: 0
NCT ‒ Doyoung: 0
NCT ‒ Hendery: 0
NCT ‒ Chenle: 0
NCT ‒ Jisung: 0
NCT ‒ Lucas: 0
NCT ‒ Taeil: 0
NCT ‒ Ten: 0
NCT ‒ Winwin: 0
NCT ‒ Xiaojun: 0
Oh My Girl ‒ Oh My Girl: 0
Oh My Girl ‒ Binnie: 0
Oh My Girl ‒ Hyojung: 0
Oh My Girl ‒ Jiho: 0
Oh My Girl ‒ Seunghee: 0
Oh My Girl ‒ YooA: 0
Red Velvet ‒ Joy: 0
Red Velvet ‒ Wendy: 0
SATURDAY ‒ Ayeon: 0
SATURDAY ‒ Haneul: 0
SATURDAY ‒ Juyeon: 0
SATURDAY ‒ Minseo: 0
Seventeen ‒ Dino: 0
Seventeen ‒ Joshua: 0
Seventeen ‒ Wonwoo: 0
Seventeen ‒ Woozi: 0
SHINee ‒ Minho: 0
StayC ‒ J: 0
StayC ‒ Seeun: 0
StayC ‒ Sieun: 0
StayC ‒ Sumin: 0
StayC ‒ Yoon: 0
Stray Kids ‒ Changbin: 0
Stray Kids ‒ Woojin: 0
Super Junior ‒ Super Junior-M: 0
Super Junior ‒ Super Junior-D&E: 0
Super Junior ‒ Super Junior-K.R.Y.: 0
Super Junior ‒ Donghae: 0
Super Junior ‒ Eunhyuk: 0
Super Junior ‒ Hangeg: 0
Super Junior ‒ Heechul: 0
Super Junior ‒ Henry: 0
Super Junior ‒ Kangin: 0
Super Junior ‒ Kibum: 0
Super Junior ‒ Kyuhyun: 0
Super Junior ‒ Leeteuk: 0
Super Junior ‒ Ryeowook: 0
Super Junior ‒ Shindong: 0
Super Junior ‒ Siwon: 0
Super Junior ‒ Sungmin: 0
Super Junior ‒ Yesung: 0
Super Junior ‒ Zhoumi: 0
T1419 ‒ T1419: 0
T1419 ‒ Gunwoo: 0
T1419 ‒ Kairi: 0
T1419 ‒ Kevin: 0
T1419 ‒ Kio: 0
T1419 ‒ Leo: 0
T1419 ‒ Noa: 0
T1419 ‒ On: 0
T1419 ‒ Sian: 0
T1419 ‒ Zero: 0
The Boyz ‒ Changmin: 0
The Boyz ‒ Eric: 0
The Boyz ‒ Jacob: 0
The Boyz ‒ Juyeon: 0
The Boyz ‒ Kevin: 0
The Boyz ‒ New: 0
The Boyz ‒ Sangyeon: 0
The Boyz ‒ Sunwoo: 0
The Boyz ‒ Haknyeon: 0
Treasure ‒ Bang Yedam: 0
Treasure ‒ Doyoung: 0
Treasure ‒ Jihoon: 0
Treasure ‒ Mashiho: 0
Treasure ‒ So Junghwan: 0
Treasure ‒ Yoon Jaehyuk: 0
Twice ‒ Chaeyoung: 0
UP10TION ‒ Bitto: 0
UP10TION ‒ Gyujin: 0
UP10TION ‒ Hwanhee: 0
UP10TION ‒ Jinhoo: 0
UP10TION ‒ Kogyeol: 0
UP10TION ‒ Kuhn: 0
UP10TION ‒ Sunyoul: 0
UP10TION ‒ Wei: 0
UP10TION ‒ Wooshin: 0
UP10TION ‒ Xiao: 0
WINNER ‒ WINNER: 0
WINNER ‒ Jinwoo: 0
WINNER ‒ Seunghoon: 0
WINNER ‒ Seungyoon: 0
WJSN ‒ WJSN: 0
WJSN ‒ Bona: 0
WJSN ‒ Dawon: 0
WJSN ‒ Dayoung: 0
WJSN ‒ Eunseo: 0
WJSN ‒ Exy: 0
WJSN ‒ Luda: 0
WJSN ‒ Mei Qi: 0
WJSN ‒ Seola: 0
WJSN ‒ Soobin: 0
WJSN ‒ Xuan Yi: 0
WJSN ‒ Yeonjung: 0
WJSN ‒ Yeoreum: 0
I suppose it's not too surprising who came out on top. What I find a bit sad is how many trophies have not been awarded at all.
