The words free and open source operating system usually bring one stock answer to mind: Linux. But Linux isn't the only FOSS OS out there, and in fact hasn't been for some time now -- it's just the one that's most immediately associated with the label.
Here's a survey of other operating systems that have also been built as open source products, are free to use, and generally come with little to no restrictions over their use. Many of them have taken strong cues from the way Linux has developed, typically in the collections of userland tools that are available. Some owe very little to Linux, apart from the general development philosophy, and are not architecturally related to Linux in any significant way.
So how useful are these operating systems? In the case of BSD and OpenSolaris, they're already in production deployments all around the world, so there's little to argue with there. HaikuOS, ReactOS, Darwin, and Singularity are entirely another story: they range from being minimally useful on the desktop to only useful as programming research projects.
What's crucial is that the work being done on all of them is openly available for any and all who want to run, deploy, adapt, transform, or build upon the work done there.
First, there was UNIX -- and then, not long after that, BSD. Derived from the UNIX mainline by University of California, Berkeley researchers in the 1970s, BSD (short for Berkeley Software Distribution) has since become a major figure in the modern operating system lineup.
Apple fans know of it as being one of the antecedents for the Apple Macintosh OS X kernel (more on that on page 4, under the section "Darwin"); embedded devices and routers, especially Juniper's network hardware, make use of it in some fashion. Our own Charles Babcock crowned BSD one of the greatest pieces of software ever written, for both the breadth of its influences and the fact that it's freely available with minimal restrictions to all who want it.
Apart from being highly polished and mature code, one of BSD's biggest advantages is its extremely liberal licensing. The source code for the whole system is freely available, and can be reused in whole or in part with little more than an attribution back to the original author. As a result, portions of BSD code show up in a great many places. Microsoft Windows itself has used bits of BSD code in its network stack, and the number of BSD derivatives, spin-offs, off-shoots, and re-workings is well into the dozens if not hundreds. Unlike Linux, though, BSD strongly retains backwards compatibility with native binaries -- in fact, BSD even offers native binary support for Linux applications.
Despite all this, BSD remains relatively anonymous in desktop settings -- apart from, again, its use as a partial cornerstone of Apple's Macintosh OS X. Most distributions of BSD are designed to be stable and dependable server systems; the emphasis isn't on desktop environments. But there are a couple of good desktop-oriented editions of BSD: PC-BSD and DesktopBSD, both using the KDE desktop system. From the outside they strongly resemble their Linux brethren, especially since most people's first experience with desktop KDE is through Linux, but that's only because BSD and Linux both share a common UNIX ancestry.
Other builds of BSD are more server-oriented, or designed for specific scenarios. DragonFly BSD's big claim to fame is powerful symmetric multiprocessing support, while FreeBSD has been developed as a single unified operating system with both system and user components developed together.