2020,
Not Wendy has some really excellent wisdom, advice, and observations so I won't repeat those; however please do reflect on them.
Thanks everyone for taking the time to write. Things have taken a turn here and I am not sure what to make of it.
I thought things with girlfriend #1 were going well. I wasn’t walking on eggshells and she was being very kind and generous to me. I felt like I was being very well looked after. Recently in our conversations, we were having increasing disagreements. Often the subject matter would be trivial, at least to me.
I was in this punk band in the mid 80’s; kind of a send up of punk in a way. Comedy-punk perhaps? We had these stage personas where we hated everything. You know, how punks invented derogatory names for themselves… Well in our recent hook up, we had been reminiscing those days and I guess I have been going into character again. Just a bit of banter I guess you’d call it. At least that is how I perceived it…
The last couple of days have been odd. I know we can’t go making assumptions, but I am getting this push/pull vibe. It is like there is suddenly this built up resentment towards me. Like my jokes have been taken personally. I admit I can be cynical and pessimistic, but
I have been painted out to be this negative
hole who likes to put everbody down. Yesterday she rang me and I thought the phone call was going well. Suddenly she declares that we are not a good match, sorry, and it is over. I was quite taken back and hurt. At the same time she is telling me she really likes me though. She is then asking me what I’ll do now, and I said I might just go back to bed for a bit. This might not have gone down well

Well most people do not like 'punks' - as you indicated they can put everybody down, even themselves with self deprecating humor. If you insulted her in this manner, take ownership for it, and apologize for it. Own your own crap.
You cannot change how you behaved in the past; however, you can change how you behave in the here and now and the future. If your partner raises an issue, even if it seems 'trivial' to you, it isn't to your partner, as they took the time and effort to bring the matter to your attention - you don't want to be dismissive towards your partner, as that will develop negative feelings towards you. At a minimum you need to validate to them that you have heard and understand their concern, and then express your opinion on it in a tactful manner if your opinion conflicts with theirs.
After apologizing for your behavior, do express you are very hurt and confused and you would like some clarification on where you might have gone wrong, listen to what she says, weigh it, and act accordingly.
Last night I am getting messages again from her. I was trying to explain (JADE-ing perhaps?) that it is unfortunate that I have not had the time to show her a different side to me; the more serious one that is loving and cares about people around me. She seems to have cherry picked some joking around comments I have made about inanimate objects and taken them very personally. I am beginning to see she is rather sensitive. Although there has been a passage of time since the 80’s, I think we are in many ways the same people we were back then, just an extension of that. She knows what my personality is like, I would have assumed.
Anyone who has been in abusive relationships can be overly sensitive to things, or poke fun of themselves with gallows humor / dark humor (I am in the 'humor' camp, myself; however, my BPD wife is not). Also have you heard the expression "Assumption is the mother of all F***ups"? There is wisdom in those words. This strongly implies that good communication is really needed here, so that there are no 'assumptions' so there is less chance of a misunderstanding that can result into the mess you now find yourself in.
Since you have admitted to being insensitive here; please take the time and effort to admit it to her, take ownership, and apologize for it. If she accepts your apology, request, to be notified if you are saying something that she finds insensitive as you are still getting to know her so you know how to adjust things as she has previously been in an abusive relationship and you want to be mindful of that. Also, please be mindful, that you have been in two very bad relationships with #2 & #3 yourself, and may have gotten BPD fleas, and have unhealthy coping mechanisms, something to unpack with your psychiatrist; however, she may be able to help you identify some of them with her feedback when you ask for an explanation.
Last night she says “thanks for fixing my washing machine” and wants to know what she should do with the shirt I left there. I told her it is old and she can toss it out. Then she is upset about that, saying a less than perfect shirt is no use to someone like me.
Perhaps you overreacted and said this in haste, and indicate this to her, in retrospect, and you will collect the the shirt the next time you see her, or she sees you. Please try not to look at this as black/white as that is what people with PD's do, but instead, shades of gray.
She tells me she needs a loving creative relationship in her life and she is now feeling down because she had to break up with her boyfriend today. I ask about these other ‘suitors’. She says she has many other men in her life but that isn’t the point. She tells me it is insane. She spends all day checking her messages waiting for me to type something. I reminded her that she said I was on trial, and unfortunately I have failed her test. She said but now she misses me. “Damned if I do etc….”
I personally think that there may have been mistakes on both sides. She had unrealistic expectations of you texting her all of the time - this is a

. Also, she said you were her boyfriend, and you thought you were still on a trial period for being her boyfriend - that is a very big 'assumption' there on her part, also a

.
Maybe I am just crap at relationships. I am going to have to spend this year looking at this in therapy. Is it possible that I picked the same partner three times, or a version of the same partner? Or am I reading too much into this? I was due to see her in a couple of days, but there is no mention of this now. And in the meantime I am receiving fluctuating messages from girlfriend #3.
No, mate, you aren't crap at relationships; but, you have had crap relationships. I think you might be reading too much into this. You have had limited experience with good relationships, so having a good relationship is a learning experience, and the only way to get better at them is to learn from your mistakes, make corrections, and do better in the future.
If you are still interested in #1, express this, that you have had nothing but crap relationships, and you want this relationship to be different, and you are seeing a psychiatrist to help you out and you aren't perfect, but it is your goal to learn and grow and be a better person for yourself and her too.
I did say I can’t really lose, earlier on in the piece. Well I guess I have a double loss now. It isn’t all bad. I get to have some alone time and to look at myself. I really don’t want a repeat of the same old…. I am not entirely convinced that #1 is finished with me yet. And that might be a worry, even a red flag.

This is not a double loss yet, it is definitely a change of dynamic where you are now behind in the score, as you indicated it is not all bad. Is this a

, maybe - it is a

, definitely, so proceed with

until it becomes a

[lose] or a
green flag [win]. There is another cliche we have here in the states "It ain't over, until it is over", so until that point is reached, there still is hope.
Reading this back…..Am I missing something here? I am, aren’t I? Is this some kind of test. I’m not all that good at this!
Right now you are likely missing something, as you have inexperience. Hindsight is 20/20 [no pun intended], once you have had a chance to reflect on this you will likely see the situation more clearly.
It could be a test. I know I have been on the receiving end of such tests, and I have also given out these tests. You could make this into a 'test', as a normal person will be able to forgive such an offense that you have done ( disparage inanimate objects ) ; however, many borderlines are less likely to forgive such an offense and will hold a grudge, or they will completely forget about it in 1 to 3 sleep cycles, as mine does some of the time.
Also, today send an apology for being insensitive with your 'punk' comments that were offensive to her, and you need some time to think about this to sort your feelings out how you could be so insensitive, and you would eventually like to get the shirt back. This will likely increase the chance of keeping the door open to #1, and also give you some time to think about it. Talk with your psychiatrist, and perhaps your sister too and move forward, in a direction that you want to move.
Perhaps, start a journal, if you haven't already (or refer to this thread) so you can see the progress that you are or are not making.
Take care, and do some self-care, whatever that might look like for you.
SD
P.S. Please keep us updated.