I have found a couple of Philips SL*s on eBay, I am wondering if I could modify it to run on my local power (120VAC) with a transfromer. I am not sure what would happen when you run a lamp meant for 50Hz AC on 60Hz AC. Would it work, or would it explode?

Magnetic ballast or electronic..? Electronic ballasts generally don't mind the frequency difference, magnetic ones can be problematic though, 50Hz ballasts on 60Hz seems to be less of a problem than 60Hz ballasts on 50Hz, which is why we ended up getting a 120V 60Hz inverter for our US stuff.

I believe you'll end up with reduced current through a choke when operating at 240v 60hz instead of the original 240v 50hz due to the frequency mismatch but at least you won't run into problems with overheating due to the core getting saturated on a higher frequency.

If the choke can take it, some people will operate it on a slightly higher voltage to account for this but it should run on 240v 60hz. If you plan to run a lot of 50Hz equipment, a 240v 50Hz inverter is probably the best way but some ballasts will require a pure sine wave model instead of the cheaper modified sine wave models. You can also get ready made frequency converters but those are usually quite costly.

Oh yes I should have mentioned that a pure sine wave inverter is a must.

Magnetic ballast or electronic..? Electronic ballasts generally don't mind the frequency difference, magnetic ones can be problematic though, 50Hz ballasts on 60Hz seems to be less of a problem than 60Hz ballasts on 50Hz, which is why we ended up getting a 120V 60Hz inverter for our US stuff.

Magnetic

I have Large British light collection down in Texas.  They will run fine on 240 volt
 60 Hz.   If u want avoid paying a buch of money on little step up transformer
  They will be perfect for the SL 18s  as I have loads of HVAC transformers that u can make auto transformers out of them.

Then at 60Hz you would need to increase the voltage a bit, to compensate for the different frequency (to about 260V or so).

as I have loads of HVAC transformers that u can make auto transformers out of them.

Where can I find such a transformer?

Me being a scrapper.   Ive been getting transformers from A/c units
That controls your thermosat on your wall.  Keep in mind these
 About all u can do with these is use them on CFLs  or small fluorescents most of these are rated at 40 VA.   So if u want such
Transformer play with small 240 volt lights.   It would cost  a
1 dollar in scrap value and postage.   U use the 120 volt input and
 Use use the over winding of transformer to get 240 volt out.
And cap off 24 volt outputs.   These transformers is the tech keep
One stock type of transformers on trucks and they also do this in industry to keep transformer part stock down.    And 120 to 240 volt
Transformers are expensive.   But what u can do is get 5 of these
And wire up all 24 volt in series and all 240 volt in parallel.   But u have to double check phasing of transformer on 240 volt side to make sure there close to zero volts when u combine them.  Or the great smoke out will result when dont.  And each bank of 5 is 200 VA
Wich is enough to power a single lamp F65T12 tube on 240 volt. 
But running british 8 foot pop pack requires two banks of 5 transformers.  U can do this for a while until u a large 120 to 240 volt
Transformer cheap.    I did that until I found a nice 120/240 volt to 240/480 volt transformer cheap at restore.  It a beefy 2000 VA
 Meaning I can run one 480 volt street light on it.   But it provided me
240 volt single phase to ground.  None of this 2 hot wire stuff to get 240 volt.