Unlockable Emblems in DJ MAX Trilogy will be available in Collection>Award>Emblem and can be equipped anytime after unlocked Not sure about the 'Runner-Up Prize' emblems though, haven't tried it myself. All these are from translations on a Japanese wiki from what I understand.

Emblem	How to Unlock
10 WINS	10 winning streak in NETWORK
10 WINS BLOCK	Blocking the opponent's 10 win streak
100 WINS BLOCK	Blocking the opponent's 100 win streak
15 WINS	15 winning streak in NETWORK
15 WINS BLOCK	Blocking the opponent's 15 win streak
20 WINS	20 winning streak in NETWORK
20 WINS BLOCK	Blocking the opponent's 20 win streak
25 WINS	25 winning streak in NETWORK
25 WINS BLOCK	Blocking the opponent's 25 win streak
30 WINS	30 winning streak in NETWORK
30 WINS BLOCK	Blocking the opponent's 30 win streak
35 WINS	35 winning streak in NETWORK
35 WINS BLOCK	Blocking the opponent's 35 win streak
40 WINS	40 winning streak in NETWORK
40 WINS BLOCK	Blocking the opponent's 40 win streak
45 WINS	45 winning streak in NETWORK
45 WINS BLOCK	Blocking the opponent's 45 win streak
5 WINS	5 winning streak in NETWORK
5 WINS BLOCK	Blocking the opponent's 5 win streak
50 WINS	50 winning streak in NETWORK
50 WINS BLOCK	Blocking the opponent's 50 win streak
55 WINS	55 winning streak in NETWORK
55 WINS BLOCK	Blocking the opponent's 55 win streak
60 WINS	60 winning streak in NETWORK
60 WINS BLOCK	Blocking the opponent's 60 win streak
65 WINS	65 winning streak in NETWORK
65 WINS BLOCK	Blocking the opponent's 65 win streak
70 WINS BLOCK	Blocking the opponent's 70 win streak
75 WINS	75 winning streak in NETWORK
75 WINS BLOCK	Blocking the opponent's 75 win streak
80 WINS	80 winning streak in NETWORK
80 WINS BLOCK	Blocking the opponent's 80 win streak
85 WINS	85 winning streak in NETWORK
85 WINS BLOCK	Blocking the opponent's 85 win streak
90 WINS	90 winning streak in NETWORK
90 WINS BLOCK	Blocking the opponent's 90 win streak
95 WINS	95 winning streak in NETWORK
95 WINS BLOCK	Blocking the opponent's 95 win streak
Basic DJ	Clear 15 Missions
Beat Maestro	Clear ALL Missions
Beginner	Clear 1 Mission
DJ Ace	5,000 play count
DJ Amateur	1,000 play count
DJ Death	Break on 4,444 combo in FREE
DJ Devil	Break on 666 combo in FREE
DJ Expert	2,500 play count
DJ Hell	Break on 851,212 combo in FREE
DJ Lucky	Break on 77,777 combo in FREE
DJ Maniac	Over 999,999 combo in FREE
DJ TR	Beginning
DJMAX Grand Master	Clear 90 Missions
Guest DJ	Clear 30 Missions
Looser	3 game loosing streak in NETWORK
Master DJ	Clear 75 Missions
Pro DJ	Clear 60 Missions
Runner-Up Prize 1	Fail to win 18 NETWORK matches as Challenger
Runner-Up Prize 2	Fail to win 36 NETWORK matches as Challenger
Runner-Up Prize 3	Fail to win 54 NETWORK matches as Challenger
Runner-Up Prize 4	Fail to win 72 NETWORK matches as Challenger
Runner-Up Prize 5	Fail to win 100 NETWORK matches as Challenger
Street DJ	Clear 45 Missions
The Lord of Battle	100 winning streak in NETWORK
The Lord of Combo	Break on 999,999 combo in FREE after completing ALL Missions