In the last two months I have experienced something new: three inner-tubes (700X25) have popped with holes immediately around the valve and on the same side of the tube as the valve. I suspect there is a deficit of support for the tube as the hole in the rim for the valve is slightly larger than necessary (1-2mm bigger in diameter that I would think best). Should I replace the rim-tape? How may the proper support be imparted to the inner-tube around the valve?
I had a spate of these; all the tubes had an elliptical, smooth rubber section immediately around the valve, and then a weld joining this to the body of the tube. The failure was always at this weld, and it looked as if the tube had herniated (bulged out in a single spot before bursting there).
After ruling out tape problems, burrs, sharps, valve hole diameter etc. - I finally figured out the problem.
The elliptical section is less flexible than the main body of the tube, so after pushing it above the bead as Zippy suggests, it wasn't fully descending to sit flush against the rim. This left a void between the rim and the less-flexible part of the tube around the valve; the more flexible section of the rim then tried to expand around the weld to fill this gap, and that's where it burst.
The solution is (touch wood) to pull the valve back down after the tyre is seated and the tube partially inflated, and tighten the nut more than usual (and then slacken off). When the tyre is fully inflated, double-check that there is no bump near the valve, that the bead is seated properly there, and that the valve doesn't move if you tug it towards the hub (not too hard!).
A picture would probably be very helpful. In particular, a close-up picture of the rim around the valve hole, with the tire installed and at least partially inflated.
I agree with the diagnosis of a Presta valve being used without a grommet in a rim drilled for a Schrader valve.
But it may be possible that more than one thing is causing your flats. When you install a new tube in a tire, the rubber holding the valve to the tube can get caught under the bead, and this can cause a flat. The remedy is to push the valve into the tire when you install it. What I mean is, push on the valve stem, like you see in the first 15-20 seconds of this video.
Too, sometimes the adhesive on rim tape is not good, due to age, oxidation, or the rim tape itself is poor quality (like the tape sold by a large national chain of bicycle stores). The tape can move around during the installation of a new tube, exposing spoke holes and/or valve holes. This can cause a flat. The best solution is to either use good rim tape, of the proper width for your rim, or, even better, to use a rim strip.
There are 3 things to try, in this order:
It can also be the result of riding the tire while the pressure is too low. When you do that the tube tends to move in the tire and can get cut against the edge of the rim.
I hope that helps.