This has been posted up before, but not in a while.
I've also googled this and found some pretty good stuff, such as this comment and this comment (both comments by the same user). I posted this up because I wanted to have somewhere to refer to from an Australian context. Hopefully we will all be able to benefit?
Understand that this may also change over time! i.e. apps that were essential a year ago are no longer essential.
Ok, I'll start..
"last minute dining" (a way for restaurants to try to sell their tables/food for discounted price last minute, rather than go unsold)
I generally don't like having an app when the web browser version of a website is better/same. For example, the mobile version of Ozbargain is great so I see no point in having an app for it. Groupon's app is not better than the mobile browser version of Groupon. AMEX app is marginally better than the web browser version.
On the other hand, Gumtree has a pretty good mobile app which I find works better than the mobile version of the website. As is the case with ING's mobile app
I'm more than open to suggestions for better apps so suggest away!