I'm a cis bisexual woman and for me pronouns on name badges/labels or emails signatures are useful for:
1) People declaring the correct pronouns for themelves - so I don't have to worry that my assumptions based on names/appearance are correct or not. I've fucked that up more than once and it's hurtful to people. It helps binary trans people whose appearance may confuse others by being clear about 'correct' pronouns. By everyone doing it, it's not JUST trans and nonbinary people being singled out.
2) Letting me know individuals within an organisation (and maybe the organisation) is supportive of trans and nonbinary people. For me that tends to correlate with bi-friendliness or at least not-biphobia.
3) Seeing who gets all het up about pronoun labels - either through ignorance or transphobia
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I put my pronouns in my work sigs.
At 2nd job where we do disability assessments within a government framework we have to link our report to the official name on a DSA funding body document. We also need to have copies of students' medical and diagnostic reports, some going back many years into their childhoods.
On our paperwork for students, we have a preferred name field so people tell us what they want to be called. Tells us who is a Thomas and who is Tom. We can also identify which people with foreign names use an Anglicised forename, which of a set of names is appropriate or which name they like to be addressed by. Over 50% of our students are "BAME".
We have a pronouns field so people can tell us how they wish to be described (report is written in third person about student). By having the pronoun field, we're already showing trans and gender variant students that we're a bit clued up and will probably be kind to them about gender stuff.
Apart from the one field using the student's funding body name, we can use whatever names and pronouns the student wants us to use in the rest of the report. The preferred name + correct pronouns fields help us work out why there is inconsistency in names/pronouns on student's documentation.
We assessors and our admin will talk sensitively with students about what names/pronouns they want in the report. Many students haven't changed their paperwork-names/gender markers for various reasons (not knowing how, family pressure etc). Some students haven't finalised name/gender-marker decisions, which is also OK (one criticism from transphobes is that gender variant young people are rushed into transition/decisions - when actually just informally using names/pronouns regardless of inconsistency is allowing choice but not forcing big decisions).
And it's not that rare either. I reckon in the 9 months I've been quality assuring several hundred reports, that I've seen between 5 and 10 clearly trans/nonbinary etc students' reports. As I have to quality assure and check assessors have got students names and personal details right, being able to see "ah, probably trans* student, that's OK" is really helpful rather than "has my colleague copied and pasted stuff with wrong pronouns". Where possible, we usually indicate the "inconsistent name/pronouns" somewhere so the funding bodies know what's going on too.
By doing this, we're providing a better, kinder and safer service for trans, nonbinary and gender variant students. We're allowing them to be who they are right now with as little pressure as possible. Our focus is their disability support, but also minimising any potential distress (which is interlinked in many cases between gender and disability stuff).
We have also, well I say we, I did the assessment and report, but colleague fought the funding body: we got a change of support provider for a trans student who wished to minimise disclosure of their trans status (which was obvious from various things) to as few organisations as possible. Student trusted Org 1 who was their named provider for one type of support, so wanted Org 1 not Org 2 for a second type of support. Initially the funding body tried to refuse saying it wasn't disability-related, but my former-lawyer colleague really fought the case for that student including that respecting their gender stuff was absolutely disability related but also the Right Thing To Do So Fucking Well Do It.
Because my team were aware trans stuff was important. Because our team did trans (and LGBT) awareness training. Because we talked to one another about "how can we be better" we could make small changes to our process and be 'alert' to things we could do to make trans people's lives a little easier... We talked anonymously about our experiences with trans students, what worked, what didn't, what we wished we'd done better in retrospect. That's like professional supervision really.
Just like we've talked about ways to make sure we don't fuck up people's "foreign names" which are initially more challenging for us to get right - but we can be better on. I showed new colleagues how to Google for name pronunciations and tricks I have for ensuring I spell things right every time (autocorrect lets you create macros, it's your friend).
For us, as disability practitioners considering 'access' as a basic right, being inclusive to trans* people is merely an extension of being a decent human being and working out 'what being decent' means to the people we basically serve. I am genuinely lucky to have such an excellent team to work with.