Punk Music - The riot grrrl sound takes influence from punk music. Punk music is usually fast and aggressive, and has lyrics about politics or issues that the singer feels strongly about, such as economic class differences. Riot Grrrl takes influence from that while incorporating feminist lyrics to create their own sound. Riot grrrls wanted to create a punk scene that was safer and more relatable to women. The punk scene was incredibly mysogynistic, and women were not encouraged to participate in it.
Grunge Music - Riot grrrl was born in the same time and place as grunge music. Like punk, the grunge scene was very masculine and unrelatable and uninviting to a lot of the women there. Kurt Cobain of Nirvana, the most famous gringe band, actually started out in the feminest punk scene that the riot grrrls were creating. It was Kathleen Hanna of Bikini Kill who came up with the famous title Smells Like Teen Spirit, referring to Kurt Cobain using his girlfriend, drummer Tobi Vail of Bikini Kills', teen spirit deodorant. The grunge scene itself takes influence from the band Bam Bam, fronted by the black woman Tina Bell, who Kurt Cobain was a roadie for.
Joan Jett and other female rock stars- Joan Jett and the Runaways were very important in women getting respect in rock music. At the time Joan Jett started playing guitar, she was told by everyone that she couldn't do it and that women didn't play guitar. She and the band she was in, the Runaways, brought all-female rock bands to the public's attention. Other important female singers at the time were Poly Styrene of X-Ray Spex and Kim Gordon of Sonic Youth. X-Ray Spex was a band in the 70s in England that helped to shape the sound of punk rock that was developing at the time. Kim Gordon was a vocalist in the band Sonic Youth and was cited as an influence and inspiration to many riot grrrl bands.