I don't know about the other alliances, but the most frustrating parts of oneworld is that many service aspects don't cross airline boundaries.

Examples:

• Calling elite lines: you can't call AA EXP line as a BA Gold, and vice versa. Getting stuck with non-elite customer "service" is not fun
• Many websites/mobile apps that require the use that FF program to get services. Example: BA mobile app (can't do much with it as a oneworld emerald), in the past CX who would not allow ti sign up for SMS notifications
• Special lines, e.g. BA's FastTrack at LHR-5, open only to its elites but not to oneworld's