If you search FT, you find a number of threads started by people who have passports which they believed suffered from minor problems and who were denied boarding by various air carriers.

Like it or not, nobody here can tell you how much fraying and of what is too much because those decisions are made by individual agents at individual locations.

You apparently hold a UK passport, but don't reside in the UK, so it is impossible for anybody here to direct you to a proper place. But, the smart move would be to obtain a clean passport and a good stiff case which preserves the passport in its original form. While I don't use my passport every day to cross a passport, I do carry it when I travel, even within the US. It is in a stiff case where nothing can get bent and nothing can be damaged.