There are 32 Trophies that can be collected.

Bronze

A Star is Born    
Completed an episode of the Game Show

Snack-Time    
Played any game outside of Game Show format

Jr. Boptologist    
Hit 10 Bop It! commands in a row without missing

Hot-Shot    
Won a game of Connect 4 Basketball against the Computer Player in under :45 seconds

Speed Shot    
Shot 10 balls in under 35 seconds in Connect 4 Basketball

Scrabble Flash Veteran    
Earned 1,500 total points (lifetime) in Scrabble Flash

Denied!    
Knocked an opponent out of the five-point zone in Sorry Sliders or were knocked out

Turkey    
Won a game of Yahtzee Bowling against the Computer Player

Silver

Celebrity    
Completed 10 episodes of the Game Show     

'A' Lister    
Completed 25 episodes of the Game Show

Snack-Pack    
Finished each of the 5 games outside of Game Show (against Computer Player)

Boptician    
Hit 20 Bop It! commands in a row without missing     

Whack-a-matic    
Hit 30 Bop It! commands in a row without missing

Bop It! Master    
Finished 25 games of Bop It!

Strong Arm    
Shot at least 100 balls, lifetime total, in all Connect 4 Basketball games     

Flawless Victory    
Won a round in 4 shots in Connect 4 Basketball

Connect 4 Basketball Master    
Finished 25 games of Connect 4 Basketball     

Quick Thinking    
Created 7 words in one game of Scrabble Five-Letter Flash     

Human Dictionary    
Created 50 unique words in Scrabble Flash (lifetime)

Scrabble Flash Master    
Finished 25 games of Scrabble Flash

Sorry… Not!    
Finished a game of Sorry Sliders where the opponent didn't earn a point     

Sorry Sliders Sumo!    
Played a game of Sumo Sliders in which two Sumo Sliders were knocked into the Sorry zone     

Sorry!    
Won a game of Sorry Sliders against the Computer Player     

Sorry Sliders Master    
Finished 25 games of Sorry Sliders

Dead Eye    
Knocked over every pin with one throw in Yahtzee Bowling Crazy Lane     

Yahtzee!    
Got Yahtzee     

Knock 'em Down    
Knocked over 600 pins in Yahtzee (lifetime total)     

Yahtzee Bowling Master    
Finished 25 games of Yahtzee Bowling

Gold

Super-Star!    
Completed 50 episodes of the Game Show

2 the Max    
Hit 30 Bop It! commands in a row without missing in Extreme Bop It!

High Score    
Scored 28 points or higher in one game of Scrabble Flash

Platinum

Family Game Night 4: The Game Show Platinum Trophy    
Achieve all trophies in the Game

http://www.ps3trophies.org/game/family-game-night-4-the-game-show/trophies/