To not let DH go to this woman’s house?

Trust is not unconditional. It is earned through trustworthy behaviour. Going alone to the home of a woman who you have said is attractive and that you might like to ask out if your wife wasn't around to have a presumably crap haircut (given that she hasn't done one before) is not trustworthy.

If he was going to try and shag her he'd do so, haircut or no haircut.

I mean you could say you're uncomfortable with him going to hers and ask him to see if she can come to yours? There's no particular reason why a hairdresser could only work from their own home. And if she's not trying to fuck him then surely she'll be fine with that?

And??? why do people say this as if it's a mic drop moment? Trust isn't something you're entitled to, you gain your spouse's trust by how responsibly you behave. This man is not thinking or acting like a married man and so quite rightly she doesnt trust him.

I wouldn’t be comfortable with this at all. As PP said, you can’t stop him but you can communicate your feelings. How did he react when you said you would prefer he didn’t go?

Please don’t try to be one of the “cool” wife's, your looking at your marriage here and he’s just looking at his ego which she is clearly stroking

He asked her because she was the one who'd mentioned it most recently. It would have made no sense to everyone else if they'd seen her make a comment, he go change, come back downstairs and then ask you if it looked better.

Honestly, I think you're making a mountain out of a molehill.

You are not being unreasonable you don’t like it (trust your gut) and he should love and respect you enough to listen to your feelings on the situation. Personally i think she sounds like she’d far too familiar with him, they are obviously closer than you’re being told. Perhaps a visit together to the pub might make you see things clearer. But stick to your guns on the no as here already admitted he finds her attractive.

This.

I wouldn’t try to stop him as if he’s going to cheat he’ll do it anyway.
But you’re clearly uncomfortable with the dynamic and if he values your relationship he’d prioritise your feelings over a freebie given he’s already said he finds her attractive.
He’s done nothing wrong yet. She messaged him and he’s shown you it. You’ve shared your views.
If she needs the practice she could also give him a truly awful haircut. Possibly more! But that on him.
See what he does next and that’ll tell you how innocent it is. Or not.

So because he changed his jacket at her comment and not yours your reaction was to ask him if he found her attractive? Did it occur to you that after the 2nd person commented he might just have thought there may be something to it after all?
You then proceed to ask the classic entrapment no win questions - do you think she’s attractive and would you go out with her if we weren’t together. To which he replied honestly.
Insecure much?
And as for “allowing him to go”. Who on earth do you think you are?
Controlling much?

So essentially you've just told your husband that you don't trust him

Who in their right mind would willingly let someone unqualified in hairdressing use them as a 'guinea pig' to practice their lack of haircutting skills on?

People tend to go to salons or hair academies on training nights, where the person training to be a hairdresser would be supervised throughout whilst they're cutting hair. If your DH is willing to let this woman loose on his hair with no supervision, then you should ask yourself why he's so keen to do it.

I wouldn't but - he has told you, he could easily avoid telling you and leave you presuming it was at a barber.

Very naive. He's not willing to turn her down himself, he'll just say "OP wouldn't let me." He knows full well it's not ok.

He hasn't exactly done anything wrong here has he?

He changed his jacket presumably because two people had now taken the piss out of it. It's not that he doesn't value your opinion but does hers, it's that having had two people tell him it looks daft in short order he'll have thought "I'm going to get this all night"

And then he answered you honestly when you asked him some stupid questions. He's not said he fancies her or wants to go out with her, he's said that she's an attractive woman who he might go out with if you didn't exist.

And then he's taken up an offer of a free haircut.

Unless there's something missing from your post OP, your jealousy seems to be the problem here, rather than anything to do with your husband

Well he asked if you would be OK with it, not if he can go. You're not OK with it, that's the truth, it's up to him now what he does with that information

He isn’t a caged animal, if he loves you he won’t do anything if he doesn’t then good riddance, stopping people doing things doesn’t last forever.

Yeah don't give him permission to cheat. He knows it's not ok.

Absolutely not, cutting hair is quite an intimate thing. She can come to yours, when you're home, if she needs to practice. If you need to tell him that, or "not let him go" something is not right. He should know it's not ok, and she is being a bit of a snake asking him.
Her making a comment about his outfit and him rushing to change it was out of order too.

Op if he has asked you I think that’s a good thing and shows he understands boundaries. If you said no and he is ok with that I can’t see the problem.

He’s an adult. He doesn’t need your permission to go anywhere, just as you don’t need his.

You don't get to give your DH permission for anything. He is an actual adult and can make his own decisions.

It also sounds from your summary that your conversation with him about her was ... potentially sub-optimal.

However, you can say "you going to this woman's house makes me feel X, and makes me think Y about how you see our relationship and that really worries me." The conversation you have about it will tell you everything you need to know about your relationship.

Or, depending on how that conversation goes and/or how things are generally, you could say "careful mate: this is dangerous territory and you may want to consider how this will impact on our marriage" (but only say that if it is actually a hard line for you and you would be willing to live up to that.)