It all depends on your interests. All the acts are very good. Some of the acts are more a fusion of styles rather than pure flamenco - some people like that kind of thing and find it more accessible than pure flamenco.

Here is the programme.
http://www.labienal.com/programa/

I should have added that many, if not most, of the names will be Google-able.

Lots of great options for these days, and here are my recommendations:

The three first days stand out for dance performances, and the two last for song.

Dance:
on Sept 7 and 8, native sevillana María Pagés (the "dancer with the endless arms") dances her latest performance Utopia in the huge and modern Maestranza theater. This will be one of the performances that draws most audiences during this Bienal, and you need to be quick to get tickets. Some tickets use to be released later in the summer if thay at first seems to be sold out. http://www.mariapages.com/en_index.php

On the 8th, up and coming Olga Pericet dances in the wonderful Lope de Vega theater. She has been a runner up to the title of best female flamenco dancer for a couple of years now. Here together with my favourite dancer on the planet, Rocío Molina, in rehearsals for the Spanish National Ballet: http://www.youtube.com/watch?v=O0gUCC5yFkg

On the 9th, the world's most brilliant and cutting edge flamenco dancer Israel Galván, also from Sevilla, dances "Sólo" in the Cartuja Monastery. This is a very special performance, he's alone and it's all about dance, no music, no song, and only some 35 mins. I saw this a couple of years ago, and it's one of the strongest flamenco nights I've experienced (and those are many for the past ten years). A friend said she had never excperienced goose bumps in her face before she saw this performance...
Galván in "Sólo": http://vimeo.com/18742537

Song:
Sept 10: I consider Marina Heredia to be the greatest female flamenco singer at the moment, at least in the younger generation. She will fill the huge Maestranza theater, this can be a truly magical night: http://www.youtube.com/watch?v=FkkI9q2viWA

Sept 11: Young and fast rising star Rocío Márquez sings in Espacio Santa Clara early in the evening. http://www.rociomarquez.com/

Jesús Méndez, one of the best young male flamenco singers on the current scene, sings in the same place at 11pm. Here together with the before mentioned Rocío Molina recently (Méndez comes in after a couple of minutes): http://vimeo.com/37918925

Should I be pushed to choose two performances on your dates, I would go for Israel Galván and Marina Heredia.

TonyBishop - Thanks for your input and posting the link

kimhe - Thanks so much for your recommendations. My son and I are trying to make up our minds. He is particularly interested in seeing Israel Galvan. Is the Cartuja Monastery easy to find? I noticed the show starts at 2300. Will I find a cab when the show is over?

sorry to interrupt OP post,

Kimhe, any recommendations on Flamenco at Sevilla from August 5 to 7.

Many Thanks.

I see the Israel Galván performance is apparantly sold out, but I'm quite sure they have some tickets left for sale that will be released later in the summer. At least I've got tickets for "sold out" performances this way in previous editions. Check on generaltickets regulararly, or else it's always possible to buy tickets outisde the venue for some few extra €.

The Cartuja Monastery is just across the Guadalquivir river from the city center, and just north of the Triana district. Will be five minutes and 6-7€ with a taxi from the Cathedral etc. Bus c1 y c2 alaso stops at Monasterio de la Cartuja. Should be no problem catching a taxi from here after the performance. http://www.labienal.com/espacios/monasterio-de-la-cartuja/

holidaying,
I come back to you

Disappointing about Israel Galvan but I will check on generaltickets as you suggest. I will next try for Maria Pages. I havent check for ticket availabity yet but can one see well from the back of the patio or is it better to try for the balcony? Is there a seating chart available for the theater? Thanks again for your help.

kimhe - I have purchased tickets for Maria Pages thanks to your recommendation. Since we probably can't see Israel Galvan can you recommend another male dancer? Thanks again.

I overlooked Andrés Marín in Teatro Lope de Vega on the 11th. He's not so unlike Israel Galván, yet he has a very personal style. In "Tuétano" (Marrow) he will be accompanied by traditional dancer Concha Vargas and great singer La Macanita.
Marín: http://www.youtube.com/watch?v=XQ-bMa55ePE
Vargas: http://vimeo.com/15469240
La Macanita: http://www.youtube.com/watch?v=mbWQJy2T7U8

Lope de Vega is my favourite theater in Sevilla, close to the Cathedral and next door to the old tobacco factory (now the university). http://www.teatrolopedevega.org/

Still tickets left.

Could be a great final night!

Note! Andrés Marín won two of the prestigious Giraldillo awards at the last Bienal in 2010, also the one for the entire festivals "Magic moment" for his dance duet with Concha Vargas. She's with him this time as well...
http://www.andresmarin.es/index_EN.html

holidaying,

I can't find any theater/festival performances in Sevilla for your dates, but they might very well turn up on this detailed and always updated agenda/program site: http://www.guiaflama.com/agenda/

You can also see the real deal every night at Casa de la Memória in the midst of the Santa Cruz district. Popular with tourists, but intimate setting and only great artists perform here. Changing artists every night, but the program is not set yet for August. The week before, fabulous Pastora Galván dances Sunday, Wednesday and Saturday. She was voted best female flamenco dancer by Spanish critics last year, and maybe you're lucky. Here is Pastora Galván together with her singer husband Cristian Guerrero. They work together in Casa de la Memória as well: http://www.youtube.com/watch?v=k2SBUcrJgeQ

But the other artists are also great. Adela Campallo, Rafael Campallo and La Choni performs here regularly, and they have all been major solo performers in the most prestigious theaters and in the greates festivals. Performing at Casa de la Memória means you're up there.
http://www.tripadvisor.com/Attraction_Review-g187443-d539294-Reviews-Casa_de_la_Memoria_de_Al_Andalus-Seville_Province_of_Seville_Andalucia.html

Thanks kimhe

kimhe - I just purchased tickets for Andres Marin. Thank you again for your advise. My son and I are eagerly anticipating both shows.