How to find the geometry of singlet/triplet states?

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I am completely lost with this question...

for example for 2 electrons coupling that results in a singlet state with a spin momentum of 0 or a triplet state with spin momentum √2h-bar. How would you go to find the geometry of these states?
 
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I would guess start by finding the probability density. See what happens when the two electrons occupy the same location in the two different states.
 

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