Alicia is a founding member of The Klezmatics and of the all-women band Mikveh. She is considered by many to be the one of the best klezmer (literally meaning "vessel of music") fiddlers in the world.

During the past decade, Alicia revived klezmer fiddle playing, which came close to extinction in this century; traditional klezmer violin style is now being played again by hundreds of her students, including most of today's best professional players. She taught and toured with violinist Itzhak Perlman, who recorded her compositions as duets with Alicia accompanied by the Klezmatics, and she was awarded first prize at the Safed, Israel international klezmer festival competition.

In Alicia's band the Klezmatics, she and five other musicians created contemporary Jewish roots music that combined the joyous and mystical Yiddish folk tradition with a postmodern aesthetic and an overtly political world view. For sixteen years, Svigals toured internationally with the Klezmatics and recorded five albums which reached the top ten of the Billboard, College Music Journal, and European World Music Charts.

In 2003 a lawsuit between Alicia and the Klezmatics was settled out of court (she claimed she had been discriminated against by being fired from the Klezmatics after giving birth to a child and that the Klezmatics had decided that this made her insufficently available for touring).

Alicia leads lectures and workshops on Jewish music for adults and children, directs klezmer string orchestras, loves to play weddings and bar and bas mitzvahs, and has published essays on Jewish music and culture.