If only traditional heterosexual relationships are acknowledged, this is the only type of love triangle that can exist. Unfortunately, this type of triangle is pretty dull. Only so much drama can be milked out of, "Will Joe chose Mary or Sue?" Usually either the answer is obvious, or the chooser quickly begins to look extremely indecisive.
Fandom's version of this type of love triangle can get pretty annoying, too. There you get old classics like: "Buffy never really loved Riley and here is all the evidence that Spike is the one true person most suited to be her partner". The basis of the love triangle itself can be unclear: If the question is, "which person does the chooser really love?", it raises a lot of other question. For the one the chooser doesn’t love, how can he/she be a rival when the chooser wasn’t really into them? Is the chooser confused? Mistakenly convinced that they did love that person? Or was the rivalry entirely in the heads of the two competitors?
Still, sometimes these love triangles can be well done. Rick/Ilsa/Victor in Casablanca works well enough, mainly because "Who does Ilsa love more?" is not the most significant question. Obviously she loves Rick more - but she ends up with Victor because there is a lot more at stake with this love triangle then the problems of three little people. And she was married to Victor before she ever met Rick, complicating even further.
I can only think of one unambiguously amazing example of this type of love triangle - and it only lasted an episode. Tara/Willow/Oz in New Moon Rising. It is significant, I think, that this is not a heteronormative love triangle (“You mean Tara has a crush on Oz? No. Oh!”). It’s also helpful that Oz and Tara are never really in direct competition. Willow has a deep relationship with Oz. He leaves. She’s devastated, then gets into a deep relationship with Tara. Oz comes back. It’s not that she truly loves Tara and isn’t that into Oz. It’s that she can’t go back to the person she used to be, and she won’t walk out on her commitment to Tara just because Oz chose this moment to come back.
If we step into the real world, there's also the Beatles break-up triangle of Paul McCartney/John Lennon/Yoko Ono. Since this one happened in the endlessly complicated real world, you could even throw in other parties, such as Paul's wife Linda or one of the other two Beatles, to add extra points to the triangle-turned-polygon. But at its most basic, McCartney/Lennon/Ono love triangle is striking because not only is it very much not heteronormative, but it is defined by John and Yoko's challenges to the rules about gender that defined their world. Yoko usurped Cynthia as John's domestic partner, and usurped Paul as his artistic partner - defying the ban on female people in the studio. It turned out that for John, on an emotional as well as career level, supplanting Paul was a bigger deal than supplanting Cynthia. The men he toured with and worked with were more emotionally important to him then the women he slept with, the woman he established a home with, the woman who raised his son. As such, the love triangle (and the hints that, had things gone different, there could have been some kind of emotional threesome) between Lennon and Ono and McCartney is fascinating to me.
Scott/Jean/Logan in Marvel Comics is a fictional traditional love triangle that works surprisingly well. Part of it may be that Logan rarely feels like a serious threat to Scott/Jean - and that Logan does gradually develop strong connections with both Scott and Jean.
But overall, traditional love triangles are pretty dull. Whether it's Fred torn between Gunn and Wesley, or torn between Wesley and Knox, or whether it's Peter Parker torn between Gwen Stacey and Mary Jane Watson, they just aren't that compelling to me.
But a love triangle does not need to be traditional.
I can think of a couple of BtVS examples of non-traditional love triangles. One is Faith's attempted seduction of Angel in Enemies. At first glance it looks like a traditional triangle, with Buffy and Faith competing for Angel. But in fact, it is Buffy’s love that Faith craves, and she goes after Angel – and later Riley – because she is envious of the affection Buffy gives them (“She was over us a long time ago Joyce. Too busy climbing onto her new boytoy to give a single thought to the people that matter.”). Faith expresses her desire for Buffy not by competing with Angel and Riley for Buffy, but by trying to take Angel and Riley away from Buffy. Very much not heteronormative.
The second example I have is completely in Willow’s head. A few episodes after New Moon Rising, Willow has a dream where Tara and Oz start flirting with each other while an entire classroom of friends, enemies, and acquantinces mocks Willow.
With these two examples, the definition of a love triangle changes. Instead of a love triangle being a competition where one choser has to figure out who they “really” love, it becomes something more complicated. After all, if you really enjoy two people…isn’t it reasonable to think that they might also enjoy each other, and possibly leave you out in the cold. And if you’re jealous of someone’s partner and insecure about being able to win that someone’s love…maybe you can at least lure their partner away and let them feel as rejected and unloved as they make you feel.