A quick search suggests that status quos is most common pluralization of status quo. This form, however, is deeply unsatisfying. Clearly, status is the noun in this phrase, while quo is some sort of adverb or something. Statuses quo seems to me the most natural attempt to pluralize (but perhaps a different form of quo is necessary; I am not a Latin expert). Which plural form(s) are correct? Can I use statuses quo?
Here is an Ngram chart for the years 1900–2008 tracking "status quos" (blue line) versus "stati quo" (nonvisible red line) versus "statibus quibus" (nonvisible green line) versus "statuses quo" (nonvisible yellow line):
The red line, the green line, and the yellow line aren't visible because the matches for those terms are too few to register against the frequency of "status quos." At least until very recently, "status quos" was virtually the only plural form of "status quo" in use in English. I say "until recently" because of the fourteen verifiable Google Books matches for "stati quo" since 1943 (when the oldest match appears), seven are from the years 2000–2013; and of the fifteen matches for "statuses quo" since 1968 (the oldest), ten are from the years 1998–2015. So those two forms may be improving on their past very low level of popularity, though not by leaps and bounds. In contrast, statibus quibus appears in just two Google Books matches of English texts, one from 1836 and one from 1959.
A Google Books search finds matches for "status quos" going back to 1909. From William Bates, "The Japanese Our Coming Merchants and Carriers: The Apparent Intention of the "Status Quo" Agreement with Japan" (1909):
No President can validate it ["a suspension of trade regulations, made in pursuance of a compact for 'navigation laws'"] in favor of any foreign country; and Congress should immediately repeal it, and terminate all conventions under it. The country looks to CONGRESS to step into the arena and put an end to bartering away our shipping trade, with its invaluable national advantages, for "open doors," "status quos," or other illusive oriental benefits of limited usefulness.
There have been many instances of "status quos" in the years that followed, including 55 in the period 2006–2008 alone.
From these results it seems evident that most of the people publishing in English who are willing to hazard a plural form of "status quo" in print have adopted "status quos" as their choice, although some few have adopted "stati quo," presumably as a back formation from "status quo" via what they take to be the normal rules for creating plural forms of Latin words whose singular form ends in -us; and some few others have opted for "statuses quo," presumably on the model of "attorneys general" and "Eggs McMuffin."
Note: I didn't try to identify and count instances in which "status quo" was used as a plural form of "status quo," as discussed in comments above by Peter Shor and Janus Bahs Jacquet. I can't think of any way to separate the plural forms from the singular forms that doesn't involve examining each match for "status quo" individually, which I am not willing to do.
I needed a context. I looked at the first link provided by FumbleFingers and found this:
The HPS should consider the existing stati quo of the organisations involved (and interested) in harbour protection.
I would like it better like this:
The HPS should consider the existing status quo of the organisations involved
While a good question, I think none of the answers here provide an adequate answer.
'Status quo' does not describe an object that would be calculable. The closest translation of the term is "the existing state of affairs", or -- by simplification -- "the way things are". There is no meaningful way to pluralize the context or the meaning of this phrase, in the same way that you cannot pluralize "the existing state of affairs" or "the existing balance of relations", and so forth. Now, the status quo, however, can also be construed as either a logic or a 'state' as such. So, if you refer to politics that produce or re-produce forms of status quo, then you can say this: 'forms' of status quo. You can refer to 'status quo states' -- which while kind of redundant uses the word 'states' with a different connotation to that implied by the translation of status quo itself. A status quo state is a state in which a status quo emerges. A system of continuous change, to highlight an example, would be unable to be defined as any kind of 'status quo'. Similarly, you can refer to the logic of the status quo -- not exactly a plural form but one which can allow you to provide more nuance to your argumentation.
These concepts are explored in more detail in post-anarchism. There is generally no consistent discourse to discuss such phenomena, but 'non-hegemony' could be a good term to look up. Deleuze and Guattari also reflect on this phenomenon and have provided an account of what a logic that does not result in a status quo could be like.