There are a couple of things I'm not sure about in your question, but one thing I am sure about: that wasn't Amy Grant.

The classic, MTV-friendly '80s cop show "Miami Vice" featured a number of guest-starring appearances by popular musicians, but Christian pop star Amy Grant wasn't one of them.

One of Grant's songs was featured in an episode, but the episode wasn't about a Russian dissident. It was the season 2 episode "The Dutch Oven," in which the team chases a drug dealer, as usual. And the song wasn't "Everywhere I Go," it was "Who to Listen To," though both are from the same album (1985's "Unguarded").

The show did branch out in its subject matter in later seasons, even into international politics, but it never dealt with a Russian on the run. It came close, though, with the season 3 episode "Heroes of the Revolution," about an East German spy hiding out in Miami as he chased the man who killed his lover. It so happened that the murdered lover was Det. Gina Calabrese's (Saundra Santiago, "Gang Related") mom.

Gina was one of the show's main characters, and so the episode was mainly about her. She went undercover as a nightclub singer, which afforded a couple of scenes where the spy slipped through the nightclub crowd, but never to an Amy Grant song.

However, I will say that the look Gina adopted for these undercover performances was very similar to Grant's '80s look -- long curly hair, heavy makeup, comically large earrings (it was 1987, after all). 

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