I don't turn anyone down, even if I don't use them, it just seems lame to do so. When it comes to these world saving plots my m.o. is usually "we need all the help we can get." If I were going to do it I would need compelling reasons.
For example, if you side with the templars without having gone to Redcliffe and having met Dorian, I can see how it would come across as really convenient that this Tevinter mage shows up, says he "risked his life to get here" and tells us that the Venatori are coming. You even have the opportunity to suggest this during his recruitment dialog at Skyhold.
The Iron Bull and Sera are a couple of others that don't really come across as trustworthy or reliable initially. The best way for a spy to win their way in would certainly be to do exactly as Bull does. I can't give a compelling reason for Sera. I think reaction to her is felt in your gut rather than in your head.
When siding with the mages Cole just appears out of nowhere and is really weird about it to boot.
Vivianne almost reminds me of Morrigan and her reason for helping out: I'm a part of this world too and it certainly does not benefit me to see it, nor the institutions I depend on, destroyed. You have the opportunity to ask her all about it in the initial meeting and she presents herself very well.
I don't think there is a compelling reason to turn down Blackwall at all. From the initial meeting he seems very straight forward and appears to be what he presents himself as. I'm sure there are players that thought there was something hinky about him all along and felt vindicated after the big reveal, but it was a total surprise to me.