The Streets of Rage cast can vary in music.
Adam Hunter, Sammy "Skate" Hunter, and Blaze Fielding are most likely in to House, Dance, Techno, and R&B. Blaze is a dance instructor and her hobbies include lambada. They most likely listen to Black Box, Technotronic, Millie Vanilli, and especially Soul II Soul. Adam & Sammy would appreciate Soul II Soul the most. Sammy would also like hip-hop from the early 90s. Really, almost any one. But if I had to guest, Heavy D, Big Daddy Kane, 2Pac, Naughty By Nature, Pete Rock & CL Smooth, Public Enemy, and Snoop Dog. Axel would appreciate those songs too, but he is most likely in to grunge. Look at his fashion tastes in 4, and he's clearly behind the times. Granted he spent most the decade, out in the woods honing in his martial arts skills. Assuming 3 took place in 1994, and it's 2004 in 4, grunge was definitely dead by that point. Max Thunder would be in to Hard Rock and Heavy Metal. A similar case applies to Cherry Hunter, though she has a bigger appreciation for Jimmy Hendrix. Dr. Zan, I do not know. Given his old age, most likely rock from the 60s & 70s.
Sonic prefers rock and alternative rock, Tails is soft rock, Amy Rose loves pop and J-pop, Knuckles is rap and jazz, Rouge the Bat is soft jazz and cafe jazz, Shadow is industrial metal and dark industrial metal, and Dr. Robotnik is hard rock and techno. Blaze I guess would more so be in to classical music despite the dance heavy soundtrack in the Rush series.
Dante is obviously hard rock and heavy metal. The same applies to Trish and Lady too. Nero is more a punk rocker. Vergil is definitely a classical music person. His human side, V, makes this obvious in his violin taunts.
Leon Kennedy is straight up alt rock from the 90s and some grunge. Given his age in 1998, he most likely has a thing for 80s rock too. Jill Valentine, classical music since she can play freaking Beethoven. Maybe she's in to garage rock too and maybe some indie stuff.
Jack Cayman is definitely in to heavy metal based off his looks alone. Mad World and Anarchy Reigns both have an indie hip hop soundtrack with rock mixed in. Especially the latter, but highly doubt Jack is that in to rap. He does not hate it, but it is not his thing. Though I am sure he would love the more rock and metal type songs.