Mantis, Bugzilla, Jira, Azure devops, trac, Redmine, etc, etc, etc, etc. There are many options and "the industry" uses many of them, so I would say it's whatever works for your company\team. 

Who can log bugs?

Just the developers or is it open for the users or public to log bugs.

For in house only stuff my team use Jira https://www.atlassian.com/software/jira

But we also use a few other Atlassian products / services so using Jira makes sense for us.

I have used bugzilla and jira and I prefer the latter.

Mostly because of the integration with other devops infrastructure such as gitlab and because of the superior ability of jira when it comes to inter-team working and collaboration.

attewell: Just feels bit dated. Still works fine.

gzt:

attewell: Just feels bit dated. Still works fine.

Are they actually using the latest version?

Yeah latest version is 5.0.6.. which was released 2 years ago, and was only tweak from release the year earlier

Spiceworks costs as much as Bugzilla. Admittedly, it's more IT Helpdesk than software development.

Or you can have a look here: https://alternativeto.net/software/bugzilla/

Jira works quite well.