Check this link out. In a short, your corolla has engine problem, like mine. Toyota is aware of it and sent out this bulletin a long time ago. I don't know why dealers have no clue and keep us running around. I finally found and printed out this bulletin and took to the dealer. Good luck.
Interesting! I notice a half-second-long "chirp" from my '09 S model, but thought it was only early-morning belt slippage. It doesn't happen when the car has warmed up. I will listen for it next time, and check my VIN. Thanks for the TSB link!!
UPDATE: Printed out a copy of the T-SB and walked it in to my original dealer, Superior Toyota in Parkersburg, WV. The curly-headed blond lady behind the service desk couldn't have cared less about it. As if to tell me to "shove it", she asked my name and proceeded to tell me, after a brief scan at her computer monitor, that she didn't have my VIN on file. Can you imagine? The dealer from whom you purchased your Toyota HAS to keep your VIN on file, right? Right. So, I took the bulletin to my good friends at C & C Toyota Scion in Marietta, Ohio, and was told, in essence, "no problem, we'll take care of it". Now, THAT is how to be of service. I was also told they'd have to check it first thing some morning, which meant I'd have to leave it with them. As it turned out, that was going to be a problem, as they were so much further down the road than Superior.
So, the next day, as I was trying to figure out a way to get my 'Rolla to them, the noise disappeared. It did not appear the next day, either. Several days passed before the "chirp" reappeared. Seems it only occurs every 6-7 starts, so, I'm going to leave well enough alone.