Just a little over 1 second. And that 1 second is actually the camera app's opening animation, so basically you can hit that Shutter button almost instantly as it appears. Its that fast.
Haven't tried burst mode (if there's any), but for Portrait mode, there's slight delay when the phone is saving the image before you can take another shot.
And in NightScape mode, you need to hold the phone for 2-3 seconds before moving away from the subject, and another 2-4 seconds to wait the image saving process. The result really worth the wait, though.
Update: Burst Mode works by holding the Shutter button, and the app will count from 1-10 (within less than 2 seconds) when taking images. After 10 shots, the Burst will automatically stops. In Gallery app, there will be icon denoting that an image is from Burst Mode. When you tap it, you can choose 1 or some or all photos to be saved.