1. While we were on holiday, we were spending most of out time sightseeing. Mistake: "out" should be corrected to "our". Corrected sentence: While we were on holiday, we were spending most of our time sightseeing.

2. Paul listened to his iPod, so he did not hear the doorbell ring. This sentence is grammatically correct and does not contain any mistakes.

1. While we were on holiday, we were spending most of our time sightseeing.⇒ spent

2. Paul listened to his iPod, so he didn't hear the doorbell ring.⇒was listening