Oddly enough, the only returning opponent that Godzilla had during the Heisei series was Miki Saegusa. Here's a rundown of the love/hate relationship Miki has had with Godzilla over the years.
Miki first locked horns with Godzilla in Godzilla vs Biollante. Due to a particularly bad guess by the Japanese military, the entire Japanese Defense Force was off guarding a different part of Japan when Godzilla surfaced near Osaka.
The only person who was close enough to the area to try and stop Godzilla was Miki Saegusa. Flown out to an unfinished section of the Kansai International Airport, Miki faced down Godzilla.
Using her psychic powers [which turned Godzilla a rather funky shade of orange] Miki tried to push Godzilla back out to sea, or at least stop him from attacking a nearby nuclear power plant.
Miki's down! Having pushed her powers beyond their limits, Miki collapsed exhausted. While she managed to keep Godzilla from attacking the nuclear plant, he ended up lumbering straight into downtown Osaka instead. Better luck next time Miki.
This was just the first time that Miki had it out with Godzilla. As soon as I get the screen captures I'll recount Miki's next spat with Godzilla. Until then, head on back to either the Miki Shrine or the Godzilla page. That, or you could head over to this section of the Miki shrine to check out a different perspective on this battle.
Update! I finally recounted the epic re-match of Miki and Godzilla. Check it out here!