How to Darn Pointe Shoes
Pointe shoes are easier to darn if they are slightly broken in, because the area under the box will have softened slightly. You'll also be able to see where you need to darn - basically, you want to darn where the satin has got dirty from contact with the floor!
Use crochet thread or embroidery thread. You'll need a strong needle with an eye that's big enough to take the thread, but a point that's fine enough to get through the satin. Suffolk sells a nice pointe shoe sewing kit that has good thread and needles all in one case.
Most "how to" guides say you should start by darning the area under the box, starting as close to the leather sole as possible. Personally, I find it too hard to get through the satin there - so I usually just darn the platform.
There are several ways to darn the platform. The conventional method involves darning into the satin on every stitch, but I've recently come across an alternative "weave" method which is well worth trying. You'll find it in the comments below, and I'd like to thank Caron Jones for letting me know about this method, and providing such an excellent description.
If you prefer to do it the traditional way, the stitches are explained in more detail below, but this video shows both stitches:
Lazy Daisy (Chain) Stitch
Using chain stitch (see pic), start at the outer edge of the platform (start a little further out than the soil mark, as that will help prevent the satin breaking away),and sew in circles. Sew all the way around until you've completely encircled the platform, and then continue in ever-smaller circles until you've covered the whole toe.
"Blanket stitch" is sometimes called whip stitch. In this case you'll start at the lower edge of the platform and sew in straight lines, overlapping slightly each time. As you move down towards the sole, remember to adjust the width of your lines so you're covering the whole platform.
You can use a combination of conventional darning and blanket stitch - start by sewing long bars of thread across the width of the toe, then blanket stitch over. Or mix chain stitch and blanket stitch, as in the video clip.
I'm sure there are other variations but those are the ones I know. You can see the two different stitches on the shoes in this photo.
I prefer the blanket stitch myself, because when I use the other methods, it can be a bit knobbly - which isn't very comfortable on the toes. But that may be because I'm so bad at sewing!
...But Do I Need to Darn My Pointe Shoes?
I always darned my pointe shoes - but then, in those days, most people did! Of course, professionals rarely darned their shoes, because they just didn't last long enough - but everyone else went through the ritual every time they bought a new pair of pointe shoes.
If you don't darn your shoes, the satin on the tip wears away very quickly and the raw edges tend to curl up, so you end up with an ugly bald-looking area around the pointe. The platform can become smushy quite quickly, too. Whereas if you darn your pointe shoes properly with crochet thread, the darning will take a long time to wear away. Yes, it will get dirty and matted, but it will stay put. So instead of a shoe with the satin peeling back from the pointe, you'll have a shoe which is pink satin right to the edges of the darn.
The darning also serves as reinforcement so the platform stays firm longer, and it will give you more traction on the floor (which is especially useful if you're doing a number with lots of pirouettes).
One compromise is to darn just the outer edge of the platform. That gives a little extra grip, and stops the satin rolling or "peeling off" at the edges, but it won't stop the platform wearing: