In general, how many points does a snowflake have?

Snowflakes always have six points.

The reason has to do with hydrogen bonding, and the way that water freezes.

Thanks to the intermolecular force of hydrogen bonding, all snowflakes have six sides, and hydrogen bonding also makes life as we know it possible. Now that’s an important bond.

Quotation from doublexscience site which is no longer available.

All snowflakes have one thing in common. They are all six-sided crystals of ice.

They can be found in seven different shapes - prismatic column, hexagonal plate, cup, hexagonal column, needle, flat plate or dendrite.

There is actually an International Snow Classification System that scientists use to observe the large variety of snowflakes.

People always say that no two snowflakes are alike. Considering the fact that the average snowflake contains 1018 molecules of water (that's 10 x 10 x 10, using eighteen 10's!), the number of combinations possible is pretty big to say the least.

https://en.wikipedia.org/wiki/Snowflake#Shape