As a general rule employees prefer fixed shifts and employers prefer rotating shifts. You can read more about the pros and cons of each approach here:  Fixed Shifts vs. Rotating Shifts.

WIth fixed shifts, you can adopt a schedule that has fixed days of work or rotating days of work. A fixed-fixed schedule means you work the same days every week on the same shift. A fixed-rotating schedule means you always work the same shift, but the days change from week-to-week, usually following a set pattern.

Employees like the predictability of a fixed-fixed schedule. It's easy to plan the rest of your life around it. Surveys of shiftworkers found that predictability is the most important schedule feature, higher than weekends off, coverage for vacations, ability to get overtime, maximum time off, and others. You can read more about this here:  Schedule Predictability.

Fixed day schedules are frequently used in police departments and companies that operate less than 24/7, especially those that have a lot of part-time employees. With police, the officers bid on both the shift and the set of days worked. This is usually done once a year. They do this to reward seniority, since the most senior officers get the "best" schedules.

I've run into a number of medical facilities that create a new schedule every week. The most senior people have a fixed set of days they work. The scheduler then contacts everyone else to fill in the gaps. This is the absolute worst way to schedule. Not only is it a lot of work for the scheduler, but the junior employees never know when they will be working until just before the week begins.

With companies using part-time employees, fixed days makes it easier to find and schedule the part-time workers. They can work around their school schedules or other part-time jobs. They are often forced to build the entire schedule around the availability of a few key, part-time people.

The biggest problem with schedules using fixed days of work is that the employees are not treated the equally. Some people get both the shift they prefer AND every weekend off. Others (usually the newest, least-senior folks) get the worst shift and have to work every weekend. Double ouch! You can read more about this here:  Fairness.

Most of the schedules I design feature rotating days of work. The schedule follows a multi-week pattern. If you look at the 24/7 schedules examples that I sell (Schedule Examples), you'll find various options with 8 or 12-hour shifts that follow a pattern that lasts anywhere from 2 weeks to 20 weeks. When I custom design schedules, I usually match the length of the pattern to the number of employees on a shift. For example, if there are 5 people on a shift, I would create a 5-week pattern.

With rotating days of work, every employee on a shift is treated exactly the same. For example, with 5 people on an 8-hour shift, all 5 will get the same number of weekends off (maybe 1 or 2 weekends off every 5 weeks). They are all working the same pattern. It's just that they are in different weeks of that 5-week pattern. You are still rewarding seniority by allowing the most senior employees to bid on the shift they prefer. To me, this is best for everyone.