The game is basically a prequel to Red Faction. Here's a couple of the Shout Outs to the game.
The Matsuko army.
A radio ad is for a show based on the game.
A radio call in a secret lab for Dr. Parker.
The Saints make a movie which is basically Red Faction.
The Saints are something of a homage to the New Orleans Saints (both use a fleur-de-lis as their official symbol). Coincidentally, their gang colors are purple with gold accents, just like the N.O. Saints' NCAA counterparts, the LSU Tigers.
In the intro scene that serves as an advertisement for Saints Flow, Pierce clearly channels Goku when he shoots energy balls out of his hands and even does a kamehameha wave!
Except for the part where it more closely resembles Hadouken. And according to the Saints Row Wiki, the name of the weapon is "Quarter Circle Punch" in a DLC that gives you that power. At the same time, the taunt for it is vanilla and is called "Quarter Circle Forward + A".
Shaundi will swing by a coffee shop for a Boston Cream, referencing her voice actor.
One taunt in the third game belongs to John Cena. Another belongs to Ric Flair (and his music is used in the intro. Yet another is the splits performed by WWE Diva Melina. Still another taunt belongs to The Miz. The Big Bad is a wrestler, who is opposed by Hulk Hogan (whose signature taunt is also present.) He makes reference to the Montreal Screwjob and has Christian's finishing move.
In Saints Row IV, if you give the "Hadoken" taunt to Female Boss voice 1, she says "Spinning Bird Punch". A reference to Chun-Li's spinning bird kick and the fact that the voices for Female Boss voice 1 and Chun-Li are one and the same.
In Saints Row 3's Whored Mode, one of the waves on the Zombie Island map gives you a chainsaw against a bunch of zombies. The wave is called Hank East and the subtitle is "He's covered wars ya know".