Can you detect if a mutation is spontaneous or induced?
Here's a quick answer; hopefully someone will give a more complete one.  But meanwhile you've got something.  

Some genotoxic agents have predictable results.  For example, they cause Gs to substitute for Cs, or they cause mutations at specific spots on the genome.  So if you had a bunch of mutations in a single cell (or collection of related cells) that matched the modus operandus of a known genotoxic agent, then it would be reasonable to conclude that the mutations were induced.  But not all genotoxic agents are like that.  And if you only have 1 or 2 mutations, that's not enough to prove they were induced.  Also, of course, it depends on your sample size and your controls.