It depends on what kind of desktop you're talking about. It has nothing to do with design and everything with accessing and interacting with software.
Ubuntu has been making gains on the server side of things. And that's likely where Canonical, the commercial entity behind Ubuntu, will earn its profits--as it hopes to do someday.
But its initial efforts on the client side arguably are what really helped shift the limelight to Ubuntu in the first place. Ubuntu gained the reputation of being easier to install and use than other Linux distributions--factors that have kept even many open-source enthusiasts from adopting Linux on their desktops or notebooks. And user experience remains a significant focus area.
Mark Shuttleworth, who heads and financially backs Canonical, is on record with comments such as "I think the great task in front of us in the next two years is to lift the experience of the Linux desktop from something stable and usable and not pretty, to something that's art." Or more broadly, to surpass Apple, in terms of desktop experience.
I strongly suspect that there are inherent trade-offs between the flexibility and choice associated with open source, and the unified approach (epitomized by Apple) that tends to be associated with good user interface design. But the bigger issue with mainstreaming the Linux PC has nothing to do with design and everything with where we are in technology history when it comes to accessing and interacting with software.
Writers of heavyweight client applications (think Adobe Systems' Photoshop, for example) don't want to support additional operating systems. Getting the latest versions of applications for its platform is even a challenge for Apple--resurgent sales and market share notwithstanding.
While there's lots of open-source software for Linux clients, there's a very modest amount of closed-source software available. This is not especially a knock on Linux, per se--though low software costs certainly contribute to Linux's attraction in some cases--but rather reflect the decades-long winnowing of the number of platforms that software vendors are willing to support.
There's also a general maturation of the PC operating system. Linux desktop distributions, Mac OS X, and--dare I say it--Windows are far more alike than they are different. You may choose one over the other to make an ideological or stylistic statement, to gain access to specific applications, or just as a matter of personal preference. But both differences and advances are increasingly at the margins.
I think we see some of this in the relatively slow take-up of Vista. The Microsoft haters blame Vista; the blame at least equally sits on the reality that Windows XP is a good enough desktop operating system for most purposes.
In short, I just don't see a lot of enthusiasm for another desktop operating system in the Windows or Mac OS X mold. This is especially so because it represents the past in many ways. Many new applications are running in the network, and the client--in its myriad forms, from desktop to smartphone--is merely a portal to access them.
In a sense, this is an opportunity for Linux. In a world where all you need is a browser and some other standardized client components, why not Linux? And, indeed, I expect that we'll see Linux on a lot of thinner clients, where it will act more as the underpinning for a browser than as a more generalized operating environment.
But I think that it is important to distinguish this from Linux, the desktop OS--as that term is normally used. This isn't about running games or editing movies on the latest quad-core Intel processor. This is about powering lighter-weight clients in which the operating system--and, especially, the general application support enjoyed by any given operating system--just doesn't matter very much.