After many trials, I understood that an important reason why silvia is so tricky to use is its sensitivity to the dose.
With the stock double basket, the difference between too little coffee and too much coffee is very small.

With too little coffee, the tamp is likely to be disrupted by the rounded bottom of the basket. This results in a thin bodied, underextracted espresso.
With too much coffee, the puck has not enough headspace and its expansion is likely to be disrupted by the shower screen. This can result in two very different kinds of espressi, none of them being drinkable:
* if the puck is even enough, the espresso is overextracted
* if it's not, the stress on the puck is too high, it makes it break at the weakest points and you get massive channeling and underextraction.

All together, the variances in the dosing process, in the beans and the grind make it rather likely to get a dose outside of the acceptable range and therefore get one of these flaws, or even both at the same time in different places of the puck.

I finally bought a deeper basket. The acceptable dosing range is much bigger and it makes it much easier to get consistent shots.
I guess the added volume of the deeper basket also slows down a bit the pressure ramp up, which should help preventing channeling. I didn't measure it, it's just a feeling.
Little investment, real breakthrough!

I agree. I was baffled by bad pours and soupy 14g pucks until I realized that the naked PF I bought had come with a triple basket! Then I tried the stock double and had trouble getting any dose above 14g to work (and the 14g doses weren't easy, either.) I switched to an LM-style ridgeless double basket and it worked much better for me than the stock double. I played around with larger doses, but 16g was about the max for the ridgeless double before hitting the screw. If I'd wanted to try major updosing (19g-21g), I'm sure it would have required the triple basket.

Also, lowering Silvia's brew pressure may help with puck stability. The newer models with adjustable OPV make that relatively easy. I lowered mine from about 10 BAR to 8.5 BAR, and had fewer extraction problems.

I stupidly bought a stock Silvia double basket when I was ordering a few PF baskets to try out. My 58mm tamper binds on the ridge with a 14g dose, and anything more hits the dispersion screen because the basket is so shallow. It's completely useless with my setup.

I ordered a few of the 58mm ridgeless baskets Great Infusions carries, and they're working out really well.

http://www.greatinfusions.com/acsgroup.html

I did a little add-on to my 1st post: "or even both [flaws] at the same time in different places of the puck."
This made me have a hard time understanding what was going on.
And I guess this is why a higher end grinder like a super jolly outperforms a lower end one like a rocky: less variance in the size and/or shape of the grind means a more consistent pour, as far as I understand.
Hope this will help some people in their quest!

It's been a while. I feel an anti-Silvia rant coming on.

There's one over at Coffeegeek right now by 'Alaskan' if you desire.