Could somebody please direct me as to how I might increase my Bizarre quality?
I find myself oddly drawn to such a status.
As a guide, my levels range from 100ish (Shadowy and Dangerous) to 130ish (The other two!) and I recently find myself to be A Person Of Some Importance.
Thanks for any advice.

Bizarre (and Respectable and Dreaded) are not stats you can raise. You need to buy or otherwise gain items which will boost them. Most POSI items will help with that.

Nothing directly increases the stats bizarre, respectable or dreaded. Instead items and advanced items (like home comforts, ships, clubs, and transports) will provide some when equipped. If you have recently become a POSI, you will receive offers to obtain many such items, although all are very expensive. As far as bizarre specifically, I know that the following provide it: a zubmarine, several connection pets, the rubbery euphonium, and the young stags.[li]

edit: of course! the clay sedan chair as well.
edited by Elene on 2/18/2014

This page on the wiki will help you:

Ah, I had not realised it would be quite so costly…
Thanks for the advice!

In general, any of the items on the second page of the inventory that provide Bizarre/Dreaded/Respectable will by rather pricy.

As an aside, equipping B/D/R items reduces the required Making Waves to gained NOtability levels.