So I'm sure everyone has their own theories on this, but anyway, here's my headcanon on which monsters represent which of the seven Soul traits.

Toriel
Toriel represents Integrity. Integrity is defined as "the quality of being honest and having strong moral principles." And Toriel certainly displays this; she seems to strongly adhere to a set of moral rules and guidelines, doing everything she can to do what she believes is right- even going so far as to leave her husband (whom I headcanon as representing Kindness- but we'll get to that) because she believes what he's doing is wrong.

On top of that, in the pacifist route, she follows you to the Barrier after realizing that in leaving you alone, you would die.

And come on, if that's not enough, Sans even says she has integrity in the conversation at the MTT resort.

Sans
So I think most people already agree with this, but I think Sans is the Patience Soul. Most of the evidence for this comes from the genocide route.

Firstly, Sans doesn't engage you in battle until the end, patiently waiting and hoping that you'll change your ways.

Secondly, the Karmic Retribution attribute of his battle is like poison. Rather than spamming you with bullets and attempting to take you out with brute force, Sans takes the longer route and tries to wear your HP down to kill you.

Third is his "special attack". Doing nothing. His goal is to outwait you until you reset.

Papyrus
I think there's kinda a three-way debate about which trait Papyrus represents, between Integrity, Kindness, and Bravery. I think he represents Bravery.

For as nice as he is, I don't think Papyrus could be Kindness for a few small reasons. For one, in small situations, he's incredibly narcissistic, (thanks, Gaster XD) and that's not exactly something you would expect from a kind person. For another, in the genocide route, even as he's trying to convince you to stop, he insults you, calling you a "freaking weirdo". Again, not what you would expect from Kindness.

It's a little harder to distinguish between Bravery and Integrity. My main argument for this is because I've already ruled out Integrity as being Toriel, so it can't also be Papyrus. But there were a few other things.

First of all is the most obvious: he stands up in front of a genocidal murderer. I mean, granted, he may not know that you're a murderer as most of the neutral endings kinda imply that Sans keeps him in the dark about that kind of stuff, but still. He definitely knows that there's something creepy and dangerous going on with you, yet he still puts up a brave front and tries to stop you.

Of course, this could be seen as Integrity, as in, trying to stop wrongdoing, but I have another argument against Integrity for Papyrus.

What's one of the biggest things about Integrity? Honesty. And is Papyrus honest?

... For the most part, yes. But there are a few moments when we can see Papyrus being dishonest- never for his own gain, I might add! He's still just trying to get everyone to be friends!

Firstly is in Waterfall, with Papyrus's whole "telling Undyne what clothes you're wearing" bit, and how Papyrus inadvertently always tells Undyne what you're wearing. I don't remember the exact requirements, (I think it was if you tell him honestly what you're wearing, then change immediately after, but I'm not sure) but in one of these scenarios, Papyrus lies for you. He says something to the effects of "my friend's opinion of you is a little bit... murder-y, so instead of telling her what you told me, I told her you were wearing a [clothing item you changed into]!"

So yes, even though he feels guilty about it, you can get him to lie. Not very integral, eh?

The second time he's dishonest is, funnily enough, with Undyne again. I mean, he's not lying this time but...

Being manipulative isn't very honest, honestly.

And hoooooo boy is Papyrus manipulative.

I mean, think about it. He knows Undyne, and he uses that to get her to get along with you. After he's jumped out of the window and Undyne says she'll never be friends with you, he comes back and uses his knowledge of Undyne's competitive inability to back down from a challenge to manipulate her into getting along with you!

(This moment)

So yeah. Dishonesty goes directly against the Integrity Soul, so that can't be Papyrus, so that leaves Bravery.

Besides, what with Sans being Patience/cyan, it just fits, since then everything about the brothers is opposite.

Undyne
I know a lot of people think Undyne is Kindness because she turns you green, but I think she's Justice.

First of all, the whole colored Magic/Souls thing hardly determines which trait they represent. I mean, Sans and Papyrus both turn your Soul blue, right? So which of them has that trait? And Sans, Papyrus, and Mettaton all use blue attacks, so what about that? No, I don't think that has anything to do with Soul traits.

On top of that, Undyne hardly fits the bill for Kindness. She's rough and loud and hotheaded. I mean, would a kind person try to fight the king, just to prove her strength? Would a kind person noogie a skeleton who also happens to be one of her closest friends (end of the neutral pacifist route)? Would a kind person demand that you be friends with her?

No. Not really.

No, I think Undyne represents Justice.

I mean, think about it. Why does she hate you in the beginning? Because she thinks she's doing the just thing to free monsterkind. And why does she stop chasing you down? Because she realizes it would be unjust to persecute you when you're not hurting anyone.

Also, in her genocide battle, she literally says, "it's not just about monsters anymore, is it?" And proceeds to fight for the whole world- humans included, because she realizes that you are going to destroy everything.

And I mean, come on. Her theme is literally called Spear of JUSTICE. I dunno about you, but that seems kinda like a dead giveaway.

Alphys
I mostly think Alphys represents the Perseverance Soul by ruling it out. I've already gotten to most of the other characters, and I mentioned that I think Asgore is Kindness (again, we'll get to that in a minute) so by default, that makes Alphys Perseverance.

I was only able to find one supporting point for this, and that's in the True Lab entries. Before the amalgamates happen, she says something to the effect of "I guess I'll just keep injecting them with Determination..." and that kind of fits with the Perseverance thing of "just keep going".

I know, it's a kind of weak argument, but like I said, I just ruled everything else out for Alphys.

Asgore
So yes, I know, I've already mentioned it some number of times, now, but I think Asgore represents the Kindness Soul.

I mean, think about it. First of all, in his battle, despite the fact that he's killing you, you can practically feel the regret and sorrow at it emanating from him. And there's another thing about his battle- he doesn't want to fight you- it's clear the toll it's taking on him, what with losing his wife and taking lives- but he still does, for his people. He wants the best for his people, and that means gathering Souls- no matter the cost to him- to free the monsters. Seems pretty dang kind to me.

On top of that, pre-battle, even though he's going to fight you, he's still very nice. He admits that he wishes he could offer you a cup of tea rather than fighting. And he tries to ease your stress before fighting him by comparing the situation to another tense and scary but familiar situation, where you know you'll most likely be alright in the end- a trip to the dentist's.

And then, right before the battle, he asks if you're ready. He gives you the option to postpone the battle to go do nicer things, rather than fighting.

So yeah, I think Asgore is Kindness.

Asriel
Yes, I'm including Asriel in this list, because we've only gotten to six Souls, and technically, there are seven traits. So, you may have guessed it already, but I think Asriel represents the Determination Soul.

Now, granted, a lot of this is just speculation, but still. I like the idea.

Anyway.

For one thing, even though Chara was apparently very rough and mean, Asriel still got along with them. So maybe that's because they both shared a Soul trait?

Going off that, even though he didn't like it, he stuck with Chara's suicide plan because he was determined to help his sibling.

And then, he is the final opponent in either of the extreme routes (the two routes that require the most determination to complete.) I mean, sure, in the genocide route, you don't really "fight" Flowey, but he's still the last one to be beaten. So either way, it takes a ton of Determination to get to him and then beat him.

As far as beating him in the pacifist route, he does put up a very determined battle. You only win because you're a human, so you have a human Soul, which is stronger. A determined human is just stronger than a determined monster by default.

Like I said, though. That's just a theory.

A GAME THEO--

... wait. Dangit.

... actually. Y'know. Even if I'm not a game theorist, this is still a theory. About a game.

So,

That's just a theory.

A GAME THEORY!!

*cue the credits*