Issue tracking systems

Major issue tracking systems

Proprietary ones and/or paid services
Will not cover those, it's mainly a source of frustration to me: they tend to be all broken (perhaps not even worth listing: virtually everything that can be broken is broken at least in some of those) with no hope for improvement, yet managing to get sold and used in companies. I can go on complaining, but it won't be of much use.
Public services
Such as GitHub, SourceForge, Bitbucket, etc. Those are actually handy for hobby projects, though GitHub will probably look in a decade as SourceForget looks now: those fancy things don't seem to last long, though GitHub has plenty of apparent flaws even now: particularly in UI, and being centralized. Still, GitHub is used by plenty of rather large and nice projects as their primary BTS (and hosting, too). Perhaps it is worth highlighting Bitbucket (and Atlassian in general), where every tiny bit is terribly broken, and it'd keep spamming you if you ever [manage to] register there; extremely frustrating, and a few different feedback mechanisms are also broken.
Python, 3-clause BSD. A somewhat minimalistic system. Does not support multiple projects nicely (single-project system hacked to host more than one project), features are rather limited, not that easy to deploy properly. Though deemed good enough (and used) for large projects such as GHC.
Perl, MPL. Seems to be nice for larger projects such as GNOME, but perhaps not so nice for using in a relatively small company (or small hobby projects). User interface is not that handy. Apparently a nice and solid system overall, yet I am just avoiding it. Though as of 2020, there's a new UI full of private use unicode code points, and partially broken functionality without JS.
Ruby, GPLv2. Plenty of features out of the box, pretty nice overall. Though markup is not good, mail notifications (their headers, in particular) are a bit broken for a long time, and it is rather hard to deploy properly.
Python, MIT. Supports email seemingly well, the web interface is quite clean and simple, JS-free, has fine documentation, relatively easy to install nicely. Though apparently the documentation is outdated: pointing out that it is in Debian repositories, while it is not anymore (well, it is in oldstable), and the last release at the time of writing (2017-11-30) is almost two years old. Hence no automated security updates and awkward manual installation; risky to deploy overall.
Mostly Perl, GPLv2. Very lightweight web UI, mail-only updates, actively used and maintained for 23 years now. Same issues as with most of the BTSes though: not in repositories, rather cumbersome manual installation, configuration, and maintenance. debbugs git repository is perhaps the primary source for its documentation (and sources).
Ruby, Go, JS; MIT for the community edition. Sluggish JS-based UI with annoying animations, suggested installation method involves curl | sh, the website is full of marketing texts and broken markup with little focus on technical details, only minimal mail integration. Quite bloated, includes a CI service. Not sure why, but seems to be pretty popular at the time of writing (the end of 2018). Though it's rather a software forge, not just a BTS.
Perl, AGPL-3.0+. Similar to a mailing list archiver, but git-based: instead of server sending messages to subscribers, subscribers are supposed to pull it (via NNTP, Atom feed, WWW, or to mirror it as a git repository). I haven't tried to use it beyond reading archives produced with it, but it looks nice.
There is much more in the Wikipedia Comparison of issue-tracking systems and Comparison of project management software. Most of those are pretty bad.

Common issues

Other approaches

Common models and gateways

There are various attempts to integrate different programs and protocols (e.g., by using common models provided by RDF ontologies such as SIOC, all kinds of APIs and gateways), but they are not used any widely.

Distributed issue trackers

There are less common issue trackers that are distributed, like bugs everywhere. Perhaps worth trying, the description looks nice.

TODO files and email

That's what I'm using the most, and perhaps it is the most reliable and simple way to manage tasks. Journaling can be used together with it or as an alternative.