Paper, Spigot or Purpur: which one should you pick?

Paper for plugins. Forge for a modpack. Purpur if you already know the flags. Spigot only on an old box.

What do I install today?

Paper. Survival, SMP and minigames with plugins.

Is Purpur better?

It is Paper with more knobs. Only if you already know them.

Does Paper run a modpack?

No. Modpacks are Forge or NeoForge.

What about Spigot?

Skip it. Keep it only if an old plugin will not start on Paper.