Bug tracking and software ticketing systems are used in every shop I have ever worked with. Some of these are decidedly low-tech solutions that are no more than a spreadsheet. Some of these solutions can help you do far more than just track issues. They can even help you raise the overall quality of your processes.
Ticketing Systems Are Almost a Commodity
Bug tracking and ticketing systems are ubiquitous. A lot of companies have even built custom solutions for their internal use. The problem space is that easy to address. As mentioned before, even a spreadsheet can be sufficient for all of your bug tracking needs. I know how easy it is to build these systems from personal experience. Over the years, I have built about a dozen various solutions. The ease of designing a solution makes the market for these tools one that is highly populated. On the other hand, the applications are often reasonably priced for the functionality they provide.
The List
The list for this episode covers a couple of different technologies, open and closed source, and it has solutions that work with a variety of database engines. These can easily provide a short list for your research or check out the larger list link to find a few other options.
A good list of options: https://www.softwaretestinghelp.com/popular-bug-tracking-software/
Jira: The cloud solution is $10/month for up to 10 users and an unlimited number of projects. You can also get the on-premise version for $10/year and get free upgrades throughout that year. That version is also limited to 10 users. This is offered by Atlassian and very popular as well as powerful.
Mantis: A hosted version is available for $4.95/month (5 users but only one project) at mantishub.com. There are other hosted solutions, or you can install it for free on-premise. It is customizable but has a slightly aging interface.
Bugzilla: This is open source and uses PHP. It is highly popular with a good and active community.
Redmine – This is a Ruby-based open source solution. It is very similar to Bugzilla in look-and-feel. There is a free hosting option at http://www.hostedredmine.com/.
Zoho Bug Tracker: The free version limits you to 1 project and five users. You can step up to $40/month to support up to 25 users. It has a modern interface, good integrations, and professional features like notifications and SLA.
GitHub Issues: This is simple, free, and perfect for project specific solution. It is included with every Github project.
Bug Genie: The free version allows one project and ten users. You can move up to a paid solution for 10 pounds per month (this is limited to 5 projects and 25 users).