Oh and, just this once, I'd talk about same-sex pairings because it's kinda fun to poke fun at.
1. Ban X Mari
Okay, this is just exclusive to the movie, but this gets carried over to Magiranger vs Dekaranger. It's just a normal 'I'll definitely protect you' moment at first, but this gets more forward and proves that Ban is interested of someone of different sex. We don't know if she'd play over to Ban when he becomes the Fire Squad member, but we never know. So let's just assume she did, otherwise Ban wouldn't keep her photo around in Magiranger vs Dekaranger. This is probably the most official coupling for Ban.
2. Sen X Umeko
(Apologies for the picture. I like it better if I use human form pictures, but I can't find any!)
Okay, this is also another official coupling, and as proven in Magiranger vs Dekaranger, they ARE going out together. I find it a bit rushed, since until the Sen-Umeko episode in late Dekaranger, there isn't much chemistry between both of them, except they ride their car together most of the time. But still, that one episode was quite the powerful episode that cements them together and shows that Sen DOES care about Umeko.
3. Doggie X Swan
You can't get more official than this coupling. Doggie and Swan are like completely matching pairs, even if they're of different species. The chemistries ARE there, even though they'd deny it. First off, Doggie risks his life to save Swan from his old enemy; he kinda had a denial about being in love when asked by Miyuki from Magiranger (anyone can see that coming!) and lastly, he gets pissed when he thinks an Alienizer is stalking Swan (even though it turns out to be Tetsu it's aiming). The proofs are there, Doggie, just be more than a man and admit it!
4. Hoji X Jasmine
As Sean Akizuki said, this one is not really an official pairing, but judging from their shared past, I'd say that there IS a reason behind this pairing. Plus, it's not like Hoji was uninterested with the person with different sex, he's shown that he has interest with that one woman that he had to break up because of killing her little brother. Their professional attitudes seem to match each other too, since between the SPD, both Hoji and Jasmine are the ones who are more likely to do straightforward, professional approaches. There's not much chemistries other than that, but if you ask me what my favorite ship for Hoji is... that's with Jasmine. Okay, I may be biased because both are my favorites, but... what can I say? They look great together!
5. Tetsu X Jasmine
Now before you raise your guns or whatever to shoot me, calm down and let me say one thing: I don't bash those who likes this pairing. In a way, as Sean said, I can see why people ship Tetsu with Jasmine. After Ban left to the Fire Squad, Tetsu basically replaces him and now shares car with Jasmine (it was previously Ban's seat), and he looked kinda crushed when Jasmine pursued Babon capturing Urara/Magi Blue, and ends up captured. That's it, really. Personally, I think that's not quite enough to make a couple, being that probably Tetsu was there just to replace Ban, as well as he'd do the same if someone else gets caught. But... they're not really a couple like Sen and Umeko, just professional pair (otherwise they'd be dating like them!). It's an okay pairing, but I just prefer other ships for Jasmine (above), and Tetsu (below)
6. Tetsu X Lisa
I'd say this would be the better pair for Tetsu, since they share more past, in the same way as Hoji and Jasmine. It's not unusual that we see 'teacher-student' pairing; and Tetsu-Lisa is no exception. The newer Lisa would have an easier time to accept Tetsu as well. However, this... seems to be a rather quickly-forgotten pairing because Lisa only appeared in one episode. Still, as much as quickly forgotten this is... yeah, I think I ship Tetsu with her.
7. Ban X Hoji
Okay, this is my first yaoi pair talk. Let me make it clear that I do not ship Yaois, but I acknowledge their existence. At any rate, it's obvious that for you Yaoi fans, Ban and Hoji are the most prominent pair, especially with their usual "I'm not your partner!" shtick, as well as that episode where Hoji thought of Ban dying and ended up genuinely crying. It goes even further that Hoji gets jealous of Ban declaring Kai/Magi Red his 'partner', while he gets demoted into 'ex-partner'. Still I'd say that's more like deep friendship, since as I said before, Ban and Hoji does have interest in people with different sex (as noted above).
8. Jasmine X Umeko
And balancing out between Ban and Hoji is Jasmine and Umeko, even though Umeko eventually ends up with Sen. Still you can't deny that before Sen-Umeko came to fruitition, Umeko spends more time with Jasmine, leading to many team-ups between them, showing off girl power prominently. This is a more prominent Yuri pair, as they're paired so often, perhaps as a couple or just partners. Overall, this is one girl-girl pairing that you won't be able to miss and is highly enjoyable.
And that's all I'll talk about the pairings today. I'll see you next time when I tackle other pairing eras... And I hope you won't flatten me with a hammer because my choice of shipping contradicts with yours...