On June 30, 2012 (56108.9999884), a leap second was added to Coordinated Universal Time (UTC) at the end of the day, to keep clocks within one second of Greenwich Mean Time (GMT). 23:59:59 was followed by 23:59:60. This caused some computers to crash. Since the software should have been able to handle this, the fault is clearly a bug in the software, similar to the Y2K bug. However, it wouldn't matter if leap seconds did not exist.
I do not think that leap seconds are necessary. They only are if you want to keep UTC within one second of GMT (UT1). However, few use GMT today. In the first place, it is the true solar time only for those exactly on the meridian of Greenwich. Everywhere else has a mean solar time that differs as you go east or west of Greenwich, so before railroads every town had its own time. But today we use unified time zones, which are (mostly) one-hour offsets from UTC. The time in these zones in theory may be a half-hour more or less than actual local mean solar time, and in practice it is in many places much more than that. Plus, many places practice daylight saving time for part of the year, adding another hour. Even Great Britain is not always on GMT, and is considering advancing their clocks another hour to be in sync with the Continent. Nobody actually follows solar time anymore. So why should we care about GMT?
What would happen if we simply abandoned leap seconds is that UTC would start to drift away from Greenwich. Every few years the difference would grow by one second. Eventually, hundreds of years from now, the difference would add up to a whole hour. But so what? Local times today are already arbitrary and decided by governments. If a particular region felt that the change had progressed too much, they could simply adjust their time zone when they chose. This already happens all the time.
UT is not the only existing time scale. Various other scales have been defined based up on the same atomic second. The atomic second is based upon the ephemeris second of 1820. Since the mean solar day is gradually getting longer, and varies from year to year, the day is slightly longer now than 86,400 atomic seconds, which is why leap years are added. International Atomic Time (TAI) was synchronized with UTC in 1958 and is now 35 seconds ahead of UTC. Terrestrial [Dynamical] Time (TT) is 32.184 seconds behind TAI and was introduced in 1984 to replace and be continuous with Ephemeris Time (ET) back to 1900, so it is currently 67.184 seconds ahead of UTC. GPS time is 19 seconds ahead of TAI, and therefore currently 16 seconds ahead of UTC. These time scales do not use leap seconds, so the difference between them is constant, but the difference between each with UTC is variable. Simply eliminating leap seconds would fix UTC in relation to these time scales.
There has been discussion for the past few years about eliminating leap seconds, and this incident will likely increase pressure to do so.