AKG K240 studio vs GRADO sr60i ?

I have both, but I believe it is important what kind of music you listen to with them.

The Grado's are great with rock, especially guitar, both electric and acoustic.   Vocals are excellent...well all the mids are clear and detailed.  The fully open back gives them a different sound from the K240's...no feeling of being boxed in at all.  The bass is often maligned, but it is there, a quick punchy bass rather than a deeply extended reverberation.  What most cannot take is the hot treble, and it will wear you out after a while.  And comfort is not a great strength of these headphones.  I solved, or at least, tempered these two issues with home-made pads and the sock mod, which darkens the sound slightly and takes the edge off the treble.  Not many want to go to that trouble to adjust their sound.

The AKG K240's have better soundstage, with a broader presentation of the sound.  Mid range is also their strength, with good detail, but these headphones are more laid back.  They also have more bass, though it is not as sharply defined.  The highs are not nearly as harsh as the Grado's.  The K240's do well with movies, reasonably well with games, and they present a less colored, more analytic sound compared to the SR-60i's.  They are also comfortable, especially with the velour pads which come with the Mk II. 

So, if you want a set to have a vibrant, exciting sound, the Grado's will work well for you.  If you'd rather have a more relaxed listening experience, but still be able to hear a detailed presentation, the K240's may suit you better.  The two sets actually complement one another quite well.

The K240's will do better with an amp.  The SR-60i's don't really need one.

I definitely agree with the comments about the K240s.... They are overall pretty enjoyable, and can be listened to long periods of time without fatigued, they are one of the most laid back cans that Ive heard, bass has good texture and impact.  I would say they are a bit slow somewhat too smooth and mellow sounding for my tastes, I listen to a lot of trance, which may not be best suited to the K240.... The highs are a bit chopped off though so you might loose some detail.  While I have not heard the SR60, from reviews that Ive read, I would say they are definitely opposites.  I think the choice would come down to  the type of music that you mostly listen to.