I have used Zephyr and ALM in the past.

We previously used HP Quality Center for the last eight years. We switched to TestRail because it is more flexible and cost-effective. The price is very competitive, and even though HP is the ultimate solution, for a starter company or like in our case, the cost was a major factor. 

I learned to work with this solution, but if I had to choose a solution, it wouldn't be TestRail.
I also use ALM, and it is my preferred tool. Tosca also has the same quality level as ALM.

Our company decided to use TestRail by Gurock as a trial to monitor the dashboards for projects.

We used Zephyr before TestRail. We chose TestRail because it was already being used in other projects. 

Previously we used a customized spreadsheet tool.

Well, I really didn't want to switch, but the rest of the team did not want to go through the exercise of learning how to configure the application for our needs.

Also, the old application was unreliable. For example, I would be in the middle of writing a test case, and before I had a chance to save the test case, I would find myself back on the previous page where test scenarios are stored. When I went back into the test scenario to continue working on the test case, all the information that I had previously entered would be lost. Thus, I would have to start all over again.

I used Zephyr, but didn't make the switch. My new company was already using TR when I joined it.

No.

We used TestLink. The reason we changed was that it was rather complex to use and there was no way to indicate at what step the case failed.