I have a 03 Honda Shadow 750. Recently, it has been randomly shutting down while riding. The bike starts right up but within a mile or two the bike starts sputting like its running out of gas and it dies shortly there after. I've been able to restart it so far. The first couple times this has happened I was at about 1/4 of a tank of gas left but it also happened with 1/2 tank. The battery has been replaced as well as the spark plugs and air filter. The tank was drained and came out clean. What do I look at next?

Just had this happen to my 2007 Shadow Aero 750. Just started it to stir up the oil, so I just had the dipstick sitting in the hole. It hopped around and got grabbed. Took off the cover, and there was no visible damage anywhere other than the dipstick. I had to adjust the electrical sensor that was in there. I guess it bent it up enough to not read anymore. I bent it back closer to the toothed wheel,and tried to start the motor, fired right up. Shut her down, reassembled. Hope this helps anyone out there that was brought to this post desperately looking for answers.

try buying a new battery

After your bike stops if you take off your fuel cap dose it **** in air, if so then buy a new fuel cap and see if that helps

The button is supposed to turn off the headlight when pushed in halfway and engage the solenoid when pushed in the rest of the way. It is not supposed to engage the starter halfway but NOT all the way. I think the button is either dirty , shorting ,or damaged. You can carefully take it apart and try cleaning it . If that does not work, you will have to replace the switch assembly. New or used , either is fine as long as it works. Honda only sells the switch with the housing as an assembly.

Carb problems, ignition problems due to heat soak, and a few other things.

If it only starts when cold, and shuts off when warm, or just randomly shuts off, it's probably one of those two things.

Sounds like the battery may be weak, recharge and test, maybe replace. As far as starting, most Hondas will not start if you open the throttle any at all. Shadows being liquid cooled are generally cold natured, but you should be able to ride away with the choke on in 1 or 2 minutes, you might have to ride as far as 4 or 5 miles at highway speed before you can take the choke fully off. If it is taking longer than these times, you will need to remove and clean the carbs. the low speed jets may be partially blocked up.

They are usely on the left control or down by the fuel tank.

They could be related - but what you need to do is first clean out the oil that has collected in the air filter housing - and replace the air filter. Then, clean the oil from the engine - Simple Green and an old tooth brush works great for this. Wash the engine and surrounding areas with soap and water.
Now, with things cleaned off, you'll be able to tell where the oil is coming from after a short test ride. It could be that the oil you saw near the head gasket was actually running from the air box. Hopefully, a new air filter will cure the problems you had with low power and poor throttle response.

check the battery or possibly the bike was in gear with the kickstand down

Your kickstand switch is either broken or out of position. The bike kills the engine as a safety feature if it thinks the sidestand is down and the bike is put in gear while the engine is running.