When is my earliest opportunity to destroy a creature I couldn't counter?
Your opponent gets to put their silver on the stack first. However, you can then add your incinerate to the stack, targetting their Root Silver.


Incinerate
New Sliver


You can then pass so the Incinerate resolves, destroying the Root Sliver. Stack is now:


New Sliver


Then you cast your counterspell so the stack is


Counterspell
New Sliver


Your counterspell resolves, and the New Sliver is countered.