My Linux predictions for the next few years...

Server-side computing in 1999: Linux will continue to displace large numbers of Windows NT servers, especially in the areas already seeing heavy growth in 1998, but also expanding into new niches. Windows NT growth will continue, but more sluggishly. Claimed Windows NT installations will be increasingly overestimated due to Windows NT systems being entirely displaced by Linux servers. Traditional Unix vendors will become increasingly threatened by Linux, although they will still prefer this to Windows NT. Sun will remain in denial about Linux impacting their high-end market.
Enterprise computing in 1999: Linux will start to make inroads into enterprise computing applications, as database vendors and other key players start to come fully up-to-speed. Some companies will use Linux for mission-critical enterprise-wide applications, albeit with trepidation at first. Those who fall back on Windows NT as the presumed "safe choice" will regret this towards the end of the year when it becomes clear just how much more reliable the Linux-based enterprise applications are.
Applications in 1999: An increasing number of applications (and a few games) will be ported to Linux, hopefully including the ever-popular Quicken. However, many applications and especially games will remain unavailable, due to lack of a perceived market due to insufficient Linux penetration on the desktop. Mozilla/Netscape 5.0 (with NGLayout, XPFE & GTK+) will be released mid-year, and prove tough competition for Internet Explorer 5.0 on all platforms, including Win32.
Desktop wars in 1999: KDE will sustain a larger following than GNOME in the first half of the year, while furious development continues on each. Mozilla's use of GTK+ will give GNOME a boost mid-year; by the end of the year, it's going to be a tossup. Ideally, KDE & GNOME people will resolve their differences and seek to combine their efforts by the end of the year. Meanwhile, average users will continue using various Win32 systems in droves. Some will try Linux to see what all the buzz is about; many of these will be intimidated and give up on the idea, if GNOME/KDE aren't ready yet.
Server-side computing in 2000: In the strong areas such as web servers and file servers, Linux will approach the installed base of Windows NT while the Windows NT "growth" will go negative; Windows NT servers will be replaced with Linux in droves wherever possible. Linux will draw away an increasing share of the market now occupied by traditional Unix vendors as well. Even Sun will start to feel threatened as Linux begins to challenge Solaris in high-end SMP environments. Other traditional Unix vendors will be struggling as their market share erodes; expect at least one major Unix vendor to switch to an open-source model and seek revenue from support -- if they can figure out how to deal with licensed code they don't own. (SCO is the most likely prospect to go open-source here; this could even happen in 1999.)
Enterprise computing in 2000: Expect growing momentum for Linux-based enterprise applications in 2000 (like server-side momentum in 1999), especially as Y2K problems strike and legacy systems need to be replaced from scratch. Many companies will feel comfortable about using Linux in mission-critical applications now, and wary of using Windows NT for the same applications.
Applications in 2000: More applications will be released for Linux, including more games. The potential desktop market looks plausible now, but by no means certain. Mozilla/Netscape will once again enjoy a commanding lead over Internet Explorer, never to be seriously challenged again.
Desktop wars in 2000: The GNOME/KDE wars should have subsided by now; the new war will be the victor (as the de facto Linux desktop) against the Windows legacy. Experienced Windows users will no longer find Linux so intimidating; distributions will be easier to install than Windows itself, and the desktop will be inviting. Lack of Unix experience will no longer be a handicap. Many Windows users will begin to experiment with Linux, but still a minority of the total Windows population. Most of these systems will dual-boot between Linux and a Win32 OS.
Server-side computing in 2001: Linux will be getting the majority of this market by now; traditional Unix vendors will start to fall by the wayside if they fail to reinvent themselves somehow. Going open-source and depending on service and support for income will be the most plausible means of survival. Windows NT will be virtually dead in this market, except for diehards and legacy systems. Sun will survive, but the future of Solaris as a closed-source OS may start to look dubious, even in high-end applications.
Enterprise computing in 2001: Linux will be moving towards a majority of this market as well, including mission-critical applications. Many companies who chose Windows NT in 1999 will be undergoing the painful process of converting their enterprise applications and especially mission-critical applications to Linux rather than continue to suffer the agony of relying on Windows NT so heavily.
Applications in 2001: A groundswell of applications and games will begin to appear, spurred by the now-apparent looming desktop market penetration for Linux. More native Linux applications will appear, rather than primarily straight Win32 ports to Linux. Simultaneous Linux/Win32 releases will not be unusual.
Desktop wars in 2001: Huge numbers of Win32 users will have at least tried using Linux; many of them will using it on a regular basis. Almost all will keep their Win32 OS around and continue to dual-boot as necessary. WINE will be sufficiently advanced by this time that users may start to live in Linux and only rarely boot back into that Win32 OS.
Server-side computing in 2002: Linux will virtually own this market, much as Windows virtually owns the desktop market today. Sun may well go open-source with Solaris to hold the high-end market against Linux, which will be encroaching by now.
Enterprise computing in 2002: Linux will hold a major share of this market as well, including mission-critical applications. Few companies will risk running mission-critical applications on Windows NT.
Applications in 2002: Most applications and games will be as readily available on Linux as on Win32; native Linux development will become more common than Win32 development. Ports will start to be done from Linux to Win32. Some applications will be released for Linux and be unavailable on Win32 platforms at all.
Desktop wars in 2002: Linux will make great strides towards becoming the new de facto desktop OS; WINE will act as a drop-in replacement for required Win32 compatibility -- desktop users will be converting to Linux in droves.

Of course, these are just my opinions; certainly I could prove to be wrong about these things. A few possible show-stoppers could slow or halt the progress predicted above, such as a quick Microsoft win in the DOJ antitrust trial or major database vendors backing away from Linux support. Barring such major potential setbacks, these are my predictions for the next few years...