I use Instapaper a lot these days. I switched from Pocket a while ago for the simpler interface and because Pocket's priorities are fundamentally the same as those of Facebook: "get the user to stay in Pocket for as long as possible and throw ads at them". I much prefer Instapaper's "Read it Later" focus that works cross browser and cross platform.
As a regular user, here's a list of improvements that I would make if I had the source code for Instapaper:
The rate at which I add articles to Instapaper greatly exceeds the rate at which I read them1. The default sorting, is "Newest Saved". This means that my reading queue operates on the LIFO methodology, and I end up never reading many of my saves. My guess is that if I switched to a FIFO queue, I would be more likely to read more content. This sort option exists, but since it doesn't persist (or sync between web and iOS), there is too much friction to use it.
EDIT: It turns out that persisting sort on iOS is possible!
Make link to source more obvious
About once in every 10 articles, I want to look at the original source of my saved articles. Every time I want to do this, I have to guess how to get to it. The header portion of each Instapaper article contains the title, the date, and the domain. I never know at first glance which one will take me to the original article (it's the domain, if you're wondering), so I always end up clicking all of them. On web, these links do have a hover state, but visually distinguishing the default state would also be useful.
Bookmarklet for "I read this article"
I want to track everything I read online, but sometimes it's easier to read the article at the original source when I find it, instead of first adding it to my queue, then going to the queue to read it, and then archiving it. A bookmarklet that sends things directly to my Archive would greatly improve this flow.
Notes on public profile
Public profiles currently show a list of "Liked" articles. I rarely use the "Like" feature on Instapaper, because I rarely want to read or find an entire article again2. On the other hand, I use the highlight feature to take notes fairly often. I think being able to share these notes publicly, would encourage me to not only take more notes, but to make more conscientious notes. Additionally, I think being able to make my notes browsable directly on my profile would give my friends and family a reason to visit Instapaper and expose them to the service.
iOS app icon badge
I'm a little torn on this one. In some ways, I like that Instapaper doesn't have an icon badge indicating the length of my queue. It means that the app isn't begging for my attention by default, which is rare these days. On the other hand, although I know my queue is always going to be too long to get to "inbox zero" on any given day, it can be handy to see relative changes at a glance. I know that I'm constantly adding articles to my queue, so if I see the badge icon hover around the same number over time, I can also feel good that I'm getting my reading in also. If that number consistently increases, I know that something probably needs to change.
My reading goals are to (1) read more articles and (2) read longer articles. Exposing the number of articles I'm reading every week or month would help me improve the former and exposing the word count of each article (or, bonus: the median word count of each article in my Archive per week) would help me improve the latter.
My workflow is to scroll through Twitter in the iOS app until I drop dead, 3D Touch links and swipe up for preview, click "Share via" and tap the Instapaper icon, which I've conveniently added to the front of the Share options.
I do "like" articles occasionally when I read something so good that I would like to stumble on it again, but I never intentionally visit my liked articles. Sometimes I chance upon them and am delighted by these favorites.