The Seeds are Sprouting

You'd think at the age of 29 the excitement of seeds sprouting might have dwindled, but no, the sight of seedlings is still gloriously thrilling.

My spinach seedlings have been shooting:

 and beans:

I took the above photo this morning.  This evening when I was watering my tomatoes, I noticed that they'd grown immensely during the day and they looked like this:

 

You can use the mushroom compost lump (with the white veiney things) in the morning and first evening photo as a size reference.

Even though I've been getting a whole heap of intentional seedlings, I've also got these patches of seedlings that I'm sure I didn't sow.  The weird thing is, they're all in clumps as if I deliberately planted them.

Here's an even bigger clump surrounding my basil seedlings.  The two basil seedlings are on the top left and right of the picture (the bigger leaves):

The question is, do I rip them out now or should I wait and see what they grow into?

Part of me thinks that as the beds were full of weeds only recently they're probably just weed seeds, but what if they're self-seeding from the previous owner's vegie plants?

And what about the areas that I planted in - how do I know whether they're the intentional or unintentional seedlings?  Because I haven't grown a number of the vegies before, I don't actually know what the seedlings look like....