How careful do I have to be with magnets around my stuff?

Generally, keep 'em away from magnetic (external) media, and you’ll be safe – according to this table, most media can be erased by magnetizing field strengths of a couple of thousand Oersteds, while neodymium magnets are generally rated around something like ten thousand Oe.
If you’ve got a CRT monitor/TV, keep 'em away from those, too, not just because the metal mask might pick up magnetic fields, causing a discoloration of the image – those can usually be degaussed away --, but because they might actually deform the mask, which can’t be so easily rectified.
And you should have bought round ones, then your kid could’ve built himself a small homopolar motor!

Right. But to address Cal’s larger questions, there are generally two types of magnetic card stripe materials: high coercivity and low coercivity. Most credit and debit cards these days use high coercivity media, which requires a field strength of around 4,000 Oe to write, change or erase data while low coercivity media only needs about 300 Oe. The latter was once common in bank cards, hence the warning about magnetic wallets and other items with magnets in them. This warning does not apply to HiCo cards, and is therefor no longer applicable to the vast majority of bank cards.

Excellent. Many thanks.

Yeah I guess a large part of my question is actually “what media is magnetic, these days?” I know floppy disks are, but other than that I pretty much don’t know how this newfangled stuff works. I didn’t completely understand the table you linked to, but then I didn’t read the whole thing, either.

Just the phone that’s sitting on my desk. I kept it in a case (sold to me by the phone store) with a little magnetic clasp; I would place the phone in the case with the screen facing out (toward the clasp). After a while, it developed a little shadow in the center of the screen. Other users of similar-style phones have had issues as well, but no one has really confirmed anything.

That’s about it. Your hard drives are also magnetic media, but as I noted, they are safe from your magnets. CDs, DVDs, flash cards, thumb drives and most other forms of storage are safe from ordinary magnetic fields. At some point a magnetic field is gong to get strong enough to do some sort of damage but by the time you get to those field strength levels, you’re well out of the range of anything you’ll ever encounter.

Unless you find yourself crammed into the LHC. Then you’ve got problems.

Well that’s comforting, anyway! Thanks! I would ask how those things work if they’re not magnetic, except that first, I wouldn’t understand the answer and second… I didn’t understand how they worked magnetically, either.

Probably due to the pressure on the screen, not the magnetic field.

Reminds me of a situation from many years ago.

I was working with a bank, that was just starting to use onsite computers. They were supposed to make a backup copy each day, before sending in their data. This was done on 8-inch floppy disks.

At one bank, they showed me how they kept these backup floppies – the latest one was always kept right there on the top of the cabinet, right underneath the phone. Which was a standard Bell desk phone, with a great big magnetic coil inside to ring the bell. Had that phone rung, it would have easily messed up that backup copy. But they’d been doing this for a while, and were quite proud of how they had made sure their backup disk was readily available.