Would you believe "none of the above?" Like many others here, I use a Macintosh. (I know you weren't intending to promote Microsoft - if you were, you would have at least mentioned Pocket PC as an option alongside Palm OS.)
However, the platform is not the issue. I think you really meant to draw a distinction among hand-helds, desktops/laptops, and the Web, and listed the first platforms that came to mind. I use my desktop computer, whatever it happens to be, running a spreadsheet program, which happens to be Microsoft Excel 2001. (No, Windows users, that is not a typo.)
The spreadsheet tracks trips as planned, booked, completed, and fully or partially credited. It plotsg miles year to date on a background of straight lines representing qualification for Gold, Platinum and EXP on Dec. 31st, so I can see progress versus goals at a glance. It also calculates the positive or negative difference between YTD miles and what the required even-pace mileage would be at this point in the year for PLT and EXP.
I only track AA miles, since my trips on other carriers are infrequent enough not to need technology and I'm not likely to qualify for elite with anyone else. In the past I used the same spreadsheet for Delta.
I like this approach because I designed it to fit my preferences in terms of inputs, outputs and ease of use. (For one thing, I'm more interested in tracking progress toward status than total miles, though it does that too.) Someone else might hate it. Since I have no plans to distribute it, that's not a problem.