What are the spectator ions in #Mg(s) + Zn^(2+) + 2NO_3^(-) -> Mg^(2+) + 2NO_3^(-) + Zn(s)#?

A spectator ion is an ion that is present on both sides of a chemical equation.

In other words, in order for an ion to be a spectator ion, it must exist as such as a reactant and as a product in the balanced chemical equation.

You're dealing with a single replacement reaction in which magnesium metal, #"Mg"#, replaces the zinc cations in zinc nitrate, #"Zn"("NO"_3)_2#.

The products of this reaction are zinc metal, #"Zn"#, and aqueous magnesium nitrate*, #"Mg"("NO"_3)_2#.

The problem provides you with the complete ionic equation for this reaction

#"Mg"_ ((s)) + "Zn"_ ((aq))^(2+) + 2"NO"_ (3(aq))^(-) -> "Mg"_ ((aq))^(2+) + 2"NO"_ (3(aq))^(-) + "Zn"_ ((s))#

As you can see, the reactants' side contains two ions

• the zinc cation, #"Zn"^(2+)#
• the nitrate anion, #"NO"_3^(2-)#

Likewise, the product's side also contains two ions

• the magnesium cation, #"Mg"^(2+)#
• the nitrate anion, #"NO"_3^(-)#

Because the nitrate anions are present on both sides of the equation, they are spectator ions.

This means that you can eliminate them from the complete ionic equation

#"Mg"_ ((s)) + "Zn"_ ((aq))^(2+) + color(red)(cancel(color(black)(2"NO"_ (3(aq))^(-)))) -> "Mg"_ ((aq))^(2+) + color(red)(cancel(color(black)(2"NO"_ (3(aq))^(-)))) + "Zn"_ ((s))#

to get the net ionic equation

#"Mg"_ ((s)) + "Zn"_ ((aq))^(2+) -> "Mg"_ ((aq))^(2+) + "Zn"_ ((s))#