Strong attraction

If they are with the other person then they are 'with' the other person. Meaning that they have chosen to be with that person rather than be with you.

The fact that that person has gone back to that person and not to you is another indication that they hold that person in higher regard than you. So I wouldn't hold your breath waiting and hoping on.

Pretty people eye flirt with many people it's an ego boost and turn on for them. So I doubt it's just you who is getting it even though you think it is. It won't obviously be everyone they do it too they need to feel it's counts but I wouldn't buy too much into it. You can try to chat them up see if they might be up for it but odds are it's just an ego boost for them and nothing more. If you do nothing I very much doubt anything will come as a result of it as if the person is attractive they don't need to go around doing the asking they get plenty of interest & offers anyway.

No they are no longer with the other person - sorry if that wasn't clear.

And it's going to sound harsh but they aren't stereotypically attractive or confident, so they don't tend to attract attention from others. And it's not really flirty eye contact, it's more like we are both checking each other are there etc. It's strange, I can't really describe it?

What has restarted again? Do you mean they're dating someone else again? If so then they're not available anyway.
But if this person is single (or if they're just going on the odd date but aren't exclusive with anyone yet) then let them know how you feel or show that you're interested in them by subtle flirting and see how they respond. You never know, they could be interested in you but aren't sure what you think of them. I don't usually look at people across the room unless I'm interested in them. You may be nervous about talking to them but they might not pick up on your interest in them unless you make it more obvious. You could also try keeping in touch on social media with them a bit to hint at your interest

If they are looking at you like this it's possibly a good sign. The meaning of this probably depends on the person but I'm a shy person and not very confident and this is how I look at the person I like from across the room to check them out but often get too shy to then maintain eye contact and smile/flirt as I don't want to be too obvious in case they don't like me back.

If you think they are attracted to you, then I would try my best to be a bit bolder next time you catch eye contact. It doesn't have to be anything too obvious if you don't want others to notice. Just give a small smile, try to hold eye contact for a second longer than you usually would then look away. You could also be a bit more flirty in other ways too e.g. playful teasing, showing an interest in what they do in their free time, subtle touches (as long as you keep it respectful).

But if you really want to move things to the next step with them then ask them out. If they are attracted to you back they'll be glad you asked as they may not have had the courage to get to the next step themselves, especially if they're not a confident person. If they are not interested then if they're nice enough they will reject you politely and treat you respectfully and if not they're probably not worth it anyway.

This sounds like drama brewing, unless I misunderstood some detail. If a person is clearly attracted to you, yet decides to date someone else and potentially even continues showing signs of attraction to you while dating someone else or being in a relationship with them, I'd ditch that person and stay away from them. I'd even ditch the entire social circle if I had to. I have done that multiple times now.

Someone who is 'serious' about being interested you and has their mental state in order, ought to act in a way that aligns with their thoughts and feelings, instead of thinking one thing, feeling something else, saying something else, and also doing something else.

So in practical terms, if for instance they are 'interested' in you, then they should and try and get to know you, or give you a window of opportunity to get to know them through conversation... instead of being evasive, avoidant, dating other people, or just generally not being straightforward with you. That sort of stupid sh*t behaviour will send you on a loop. It is also insensitive on the part of such people to show signals of interest and attraction to draw you in, and then effectively reject you by going off with someone else and never having the courage to even be straightforward about it with you. That's not the kind of person that has a strong character or integrity. Nor is someone who is potentially already in a relationship, but flirting with other people. I don't like it, and I make that pretty clear with my body language/facial expression when such people act this way towards me.

That's my advice from my experience. I avoid people who I see sending mixed signals. In fact I often even avoid such people if I see them sending mixed signals to others, not necessarily just myself. This applies to guy friends too. If he's showing interest in one woman and then fks off to hook up with another... then I'm probably not going to get along with him in most social settings. For some reason that has meant that almost half my guy friends are Asian, although that's anecdotal & slightly besides the point - just an interesting detail I thought.

No, so they were dating someone else but they are now single. They didn't show interest to me whilst seeing someone else, but we did have some interactions before they were in a relationship and it's restarted now they are single.

But yes, I agree and it's the conclusion I had been coming to. It does feel messy and I don't know where I stand with them, which isn't what I want but also I feel like I don't really have any other options (as bad as it sounds).

Then you need to have a conversation with them, and they need to give you a window of opportunity to have that conversation. Clear these things up. Find out if they're actually single, or dating others, or just 'talking' to others. Get some straightforward answers.

Recently I had a woman show strong interest in me, but then avoid me any time I tried to talk to her properly one-on-one (without people eavesdropping), and yet continued to show interest. This sort of crap just annoys me. I think it would annoy anyone. So if someone does it to you, just drop them. I'm reluctant to even be friends with someone like that, not because I'm afraid of the cliched 'friendzone', but because it is annoying to deal with them on occasions repeating those same mixed signals. If they just stopped, and we could simply be friends, without the pseudo-flirtatious bs, I'd be ok with that. I don't enjoy the stupid mind games though.

Even if they were dating someone else in between, what matters is what they think of you now. As people get to know others, their feelings for them can develop and change. There was a guy that I was quite attracted to for a while but didn't think much about it at the time as I didn't consider the possibility that he could be attracted back and hadn't known him long enough/well enough to develop a proper crush on him. Then I dated someone else for a bit who had asked me out as they seemed nice before realising they weren't right for me and stopped seeing them and soon afterwards I ended up having a big crush on the first guy I mentioned the more I saw him.

I would speak to the person the next time you see them and gauge how they feel about you. Also bear in mind that if you are not obvious enough about liking them they may just think you're being friendly.