The latest Spigot and BungeeCord builds have been updated to include support for Minecraft versions 1.7.2, 1.7.5, 1.7.6, 1.7.7 and the newly released 1.7.8. In order to enable this support all you need to do is to update to at least Spigot build #1387 and BungeeCord build #883.
You will need to remove any disguise, nametag or NPC plugins. If you experience any issues disable and check all your plugins.
Additionally Spigot builds #1388 and above are full builds, not just protocol updates. Please note that these builds are still in development and quite unstable, as such staying on #1387 may be beneficial. When using these builds backups are very important as 1.7.8 makes PERMANENT changes to the way player data is stored.