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Siri can make use of the Shazam engine to identify songs it hears, which is pretty useful--especially if you're using Siri hands-free.. Unfortunately, you can't just ask Siri to show you a list of a songs you've identified. For that, you have to dive into the iTunes app or, for a more complete list, the Shazam app. Here's how it all works.
View Identified Songs that Are Available for Purchase in iTunes
The iTunes Store app has a list of all the songs that Siri has identified and also found a buying recommendation for in iTunes. If you see a Buy link next to the song when Siri identifies it, the song will show up on the Siri list in the iTunes Store app. Here's how to find the list.
Tap the "iTunes Store" app to open iTunes.
In iTunes, on the Music tab, tap the List icon in the upper right.
On the List screen, tap the Siri tab.
This tab shows a list of all the songs you've used Siri to identify and that Siri was able to find an immediate option for purchase in the iTunes store.
Unfortunately, this list doesn't show songs that you can't purchase in the store. It also doesn't show songs that Siri wasn't immediately able to link to a purchase option, even if those songs are available in the iTunes store. We haven't been able to lock down an answer on exactly when this happens, but it does seem to happen most often when more than one version of a song is available in the store. There is a workaround, though.
Collect and View a Full List of Songs With the Shazam App
Since Siri uses the Shazam engine to identify songs, it makes sense that she links up with the Shazam app if you have it installed. The problem is that linking isn't automatic. You have to send Siri's identified songs to Shazam each time to add them to your Shazam list. It's annoying, but on the plus side, at least you'll have a complete list of songs somewhere, including those not available on iTunes.
Whenever you use Siri to identify a song, you'll see a Shazam link at the bottom right. Tap that to have Siri add the song to your Shazam list. This is a particularly important step to take if Siri doesn't show a link to buy the song beside the song title. This is an indicator that the song will not show up in the Siri list in iTunes.
When you tap the Shazam link on Siri, the Shazam app opens and you can see that the song has been added to your My Shazam list.
Of course, having to tap the Shazam link to add songs isn't ideal, especially if you're in a situation where you're using Siri hands-free. But it is what it is. You'll also notice in the example above that the song "Bad Moon Rising" is clearly available from iTunes, even though Siri didn't offer the option to buy it when she identified the song.
Yes, the whole thing is a little clunky. It would be nice if Siri could automatically add songs to the Shazam list or even keep a full list of her own. But at least there's some mechanism in place for viewing the songs you've identified.