Apple and Microsoft have both suffered embarrassing starts to the new year, with iPhone alarms failing to go off and entire Hotmail accounts disappearing for several days.

Microsoft says it has now solved a problem that began on New Year’s Eve with users finding their e-mail accounts partially or completely empty. There seems to have been a mixed pattern of problems: some users had no archived messages, some had new e-mails going straight to the deleted messages folder, and some had no folders whatsoever.

According to Microsoft, it has now restored all e-mails, but is asking users who are still experiencing problems to log into the Windows Live Help site and post on a dedicated support thread.

There’s no official word yet on how the problem came about, though the most popular theory is that an important server crashed and a database needed to be restored.

The iPhone situation appears to involve iOS4 (including older devices that have had the operating system upgraded) rather than the specific iPhone 4 handset. It only affects one-off alarms rather than those set for recurring use. Apple had said the problem would be resolved by today, though there are still reports that some users are affected.

Apple staff could be forgiven for feeling they are getting some unduly harsh criticism over what is a relatively minor issue, but the real embarrassment lies in the fact that a similar problem occurred just weeks ago when the alarm feature failed to recognize the end of daylight saving time.

It’s not yet clear exactly what happened this time round, though based on previous experience among other mobile devices, I wouldn’t be surprised if one part of iOS believed there was a leap second in 2010 (an extra second added at the end of June or December to make up for the actual journey of the Earth around the Sun growing gradually slower,) even though there was none.

The problem could even be as simple as the iPhone mistakenly thinking 2011 is a leap year. A similar problem in 2009 saw one model of Microsoft’s Zune player freeze for 24 hours.

Apple isn’t alone in technical difficulties. French mobile operator Orange has received complaints of phones sending the same messages as many as 15 times. That may not be a date-related technical issue, but simply to do with the sheer number of messages sent on New Year’s Eve causing a technical malfunction.