The "star of the show" cards refers to "effects controlled by your opponent". Does that include plot twists, or just activated and payment effects?

Effects would encompass all targeted effects including plot twist and payment effects. Those effects that do not target though still will work on the character.

In the future for reference, please include the text of the plot twist with your question. Text is below.


Star of the Show - Plot Twist
Cost: 1
As an additional cost to play Star of the Show, discard an X-Statix character card. Ongoing: While you control exactly one character, it cannot be the target of effects your opponents control.

Perhaps the following from the comprehensive rules might be useful as well:

504. Effects

504.1 Triggered powers, payment powers, plot twists, delayed triggered modifiers, and the game itself can put effects on the chain. Recruiting a character or equipment also puts an effect on the chain.

504.2 The player that plays the effect is the controller of that effect. If an effect is triggered, the player who controlled the object that triggered an effect, at the time it triggered, is the controller of that effect.

Glossary:
Effect
Recruiting characters and equipment; payment and triggered powers; playing plot twist cards; and the game itself generate effects on the chain. An effect can only exist on the chain. To leave the chain, an effect must resolve or be negated.