Chart points are getting hidden by other points

Users can build charts in our app, and there's a problem: the points can overlap and sometimes they hide each other. I'd like to find a way to:

1. At least make sure user understands there is more than 1 point there
2. Select points underneath

So far I came up with such solutions, but I try to gather as many ideas as possible before we settle on one:

1. Introduce shadows to all points. 2 points in the same spot will make the shadow darker. Still no way to select 2nd point. But this option doesn't require any user input which is great.
2. Let user introduce some random noise to the data. Which will make the points spread in different directions.
3. Let user "rotate" the points as if we could look at the at an angle.
4. Introduce more shapes to the points. So that a square is visible under a circle (or vice versa). But this one has many downsides and won't always work.
5. Make points a bit transparent - this one also has lot's of problems.

Any other ideas?