About 3 years ago, I spent roughly one week reviewing the LMSs on the market at that time. I narrowed the field to 40, then down to 5. My process involved a spreadsheet with all of the features I thought we’d need, and I added some as I went through the task. I grouped these features together to make it easier, and also prioritised them. This included things like ability to customize with our company branding, compatibility with xAPI, SCORM, SSO etc, Reporting, ILT course admin, Help desk (and where in the world they are – I’m in Australia so this is important), Multilingual, Responsive, Certification, Webinar integration etc.
If I was going through the process now, I would also do some research into pricing, such as pay-per-user, pay-per-active-user, upfront license, free etc… as I think this would narrow the field quite quickly.
I wouldn’t recommend any now, since it has been so long, but as a starting point, you could identify the requirements you know you need, then research common LMS features and see if you want those as well, consider the size of your organisation and your budget and you should be able to come to a short list.