Hingis hammered Silvija Talaja of Croatia, 6-0, 6-1, to ease into the third round of the hardcourt event.
The 1998 champion, Hingis is gunning for her third WTA title of the year. The former world No. 1 triumphed in January at Sydney and last month at Tokyo, where the 21-year-old captured her 40th career trophy.
Seles, the fourth seed, cruised into the round of 16 with a 6-0, 6-3 rout of Slovakia's Martina Sucha.
After an easy first set, the nine-time Grand Slam winner was even after six games in the second before winning three straight games to close out the match.
"When you win a set 6-0, the player has nothing to lose," Seles said. "They're going to come out and whack that ball. It's a good lesson, I'll try not to repeat that."
Seles captured the Indian Wells crown in 1992. Her lone win this season came at Doha and was the 52nd of her career.
Shaughnessy enjoyed a relatively easy 6-4, 6-1 victory over Nicole Pratt of Australia. The 22-year-old sixth seed improved to 5-3 lifetime against Pratt.
Despite being a two-time winner on the WTA Tour, Shaughnessy advanced to the final 16 at Indian Wells for the first time in five attempts.
No. 5 Jelena Dokic of Yugoslavia and No. 9 Arantxa Sanchez-Vicario of Spain both needed three sets to advance Saturday.
Dokic rallied past fellow teenager Elena Bovina of Russia, 4-6, 6-3, 6-2. Sanchez-Vicario, a semifinalist here in 1995 and 1997, pulled away to beat Katarina Srebotnik of Slovenia, 6-4, 4-6, 6-1.
Meanwhile, 10th seed Amanda Coetzer of South Africa had an easier time in a 6-4, 6-4 win over French qualifier Virginie Razzano, 6-4, 6-4. Coetzer, a 1993 and 1994 finalist, is making her 12th appearance in this event.
However, 11th seed Magdalena Maleeva of Bulgaria and No. 15 Angeles Montolio of Argentina crashed out in the second round.
Danish qualifier Eva Dyrberg upset Maleeva, 6-2, 6-3, while Switzerland's Emmanuelle Gagliardi knocked off Montolio, 6-1, 6-4.
No. 13 Iroda Tulyaganova of Uzbekistan moved into the third round with a 6-3, 6-3 victory over Croatian qualifier Jelena Kstanic. Austria's Barbara Schett, the 16th seed, swept past Germany's Bianka Lamade, 6-4, 6-4.
Three other seeds fell Saturday as Spain's Marta Marrero eliminated No. 17 Francesca Schiavone of Italy, 6-4, 6-3, Anna Smashnova of Israel ousted No. 20 Rita Grande of Italy, 6-1, 7-6 (7-4), and Janette Husarova of Slovakia knocked out No. 21 Magui Serna of Spain, 6-3, 6-3.
Notable deaths of 2014 [PHOTOS]