I wouldn't call the Tosh/Owen relationship one that never happened. In fact it happened big time -- only they acted on it too late and it turned tragic.
For me, the big one was Archer/T'Pol on Enterprise. I'm almost legendary for my undying support of that show, but even I recognize a "WTF?" decision when I see it, and after spending the first two seasons setting up the relationship, and then giving us the incredible Twilight in which we find T'Pol (in an alternate timeline) dedicated her life to nursemaiding "Memento Archer", they go and suddenly create the T'Pol/Trip romance, which just never felt right after that. And for those who criticized the idea of the captain carrying on with a subordinate, remember that (in the expanded universe canon, anyway) Robert April and his chief medical officer were married, so precedent exists.
My second nomination will get some of my fellow Whovians screaming (because of there's a fandom that's more against "shipping" than Trekkies, it's Whovians), and that's the Tenth Doctor and Donna Noble. While the Doctor and Rose was a very romantic relationship (though in an unconventional sense - and at times Rose behaved towards the Doctor the way a follower behaved with a cult leader), and Martha and Jack's feelings for the Doctor were decidedly one-sided, the very fact the Doctor and Donna set out to be "just friends" actually made their relationship the more realistic, and one I could easily have seen progress into actual "romance" had they stayed together for longer. IMO the fact romance was in the cards was made evident as Journey's End reached its conclusion and Donna seemed so pleased to now be the Doctor's equal (mentally), and with the Rose situation finally resolved - and her statement that she wanted to stay with him forever ... it was pretty clear where things might have gone. In fact I'd go out on a limb to say she probably saw him from the beginning as the replacement to the fiance she lost in The Runaway Bride.