If you have a flexible calendar it is a great idea to visit Seville and surrounding towns during a Flamenco festival when the real passion of the locals and performers comes together for their own enjoyment and not as a tourist show.
The main flamenco festivals in Seville are:
Bienal de Flamenco
Flamenco during Feria de Abril
Flamenco festivals nearby Seville
Every July, many visitors to Seville make a day trip to Cordoba which hosts a guitar and jazz festival, which also features a lot of Flamenco. There are local flamenco festivals in Moron de la Frontera (August - Gazpacho Andaluz), and Utrera (June - Potaje Gitano) - both cities are less than an hour from Seville. And the nearby town of Arahal hosts a festival for Flamenco in June of every year.