Step 1: Understand what an oxidation-reduction (redox) reaction is. A redox reaction involves the transfer of electrons between species, resulting in changes in their oxidation states.
Step 2: Analyze each reaction by assigning oxidation numbers to all elements on both sides of the equation to identify if any element changes its oxidation state.
Step 3: For the reaction \(\mathrm{Zn + Cu^{2+} \rightarrow Zn^{2+} + Cu}\), note that Zn goes from 0 to +2 and Cu goes from +2 to 0, indicating electron transfer and a redox reaction.
Step 4: For the reaction \(\mathrm{Fe^{2+} + Cl_2 \rightarrow Fe^{3+} + 2Cl^{-}}\), Fe changes from +2 to +3 and Cl changes from 0 to -1, showing electron transfer and a redox reaction.
Step 5: For the reaction \(\mathrm{NaCl (aq) + AgNO_3 (aq) \rightarrow AgCl (s) + NaNO_3 (aq)}\), check oxidation states: Na, Cl, Ag, and N remain the same before and after the reaction, indicating no change in oxidation states and thus this is NOT a redox reaction.