Why might my inner tubes be getting consistently punctured? I recently had a flat and replaced the inner tube. It pumps up just fine, but then when I leave it and come back it's completely flat. I replaced the tube twice more, with the same result each time. 

The the punctures were


Just to the right of the valve on the rim-side, not a clean cut but a small hole as if the material had been stretched or twisted too far.
Roughly opposite the valve, a 6 or 7 inch gash, not straight. I couldn't tell whether this was rim-side or tyre-side.
Just to the left of the valve, rim-side, a clean hole about 5mm long.
edit: replaced rim tape, put in new inner tube. 2 minutes later I hear it go pssshhh, same result as number 3.
edit: added second new rimtape on top of first, new tube, 2 minutes, pssshhh, same result as number 4.


I'm using the process described in this video, except that I don't use a tyre lever to put the tyre back on after inserting the new tube (because I've previously made the mistake of pinching the inner tube and making a hole), instead I just use my thumbs to push the bead over the rim.

I've examined the inside of the tyre and the rim, I haven't found any aberrations (no spokes sticking through rim tape, no holes or foreign objects anywhere). Edit: chap in repair shop had a look at the tyre and confirmed it seems fine.

Could these different failures all be caused by the same thing?
So, I think the problem is that I have a 25mm tyre on a rim designed for narrower tyres. That means the rubber surrounding the valve is distorted into the narrow valley of the rim, which puts stress on it. As @Batman suggested, failure #2 seems to be a red herring.

So, I've opted for the narrowest tube available and furthermore used a cutting from an old tube to provide padding around the valve, like this, so that the rubber doesn't have to distort so much to reach the base of the rim valley.

It's still inflated so far, will mark this solved if it's still inflated at the end of the day!