I've been trying to look this information up but can't seem to find a list for all of the girls, ponies not EQG. Here are the ships I know about.
-Twilight: Flash Sentry.
-Pinkie Pie: Cheese Sandwich, Pokey Pierce, Party Favor
-Applejack: Caramel.
-Rarity: Fancy Pants.
-Fluttershy: ??
Rainbow Dash: Soarin.

Does anyone know any others?

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Fluttershy: Big Mac
Fluttershy: Discord

Two of the more common pairings for her (that I've seen).

Twilight/Soarin pops up now and then (or used to before Flash entered the picture)

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that one balloon pony from Our town. (I can't for the live of me find his name)

Party Favor?

6625041
I didn't include Big Mac mainly because he has a canon ship now, but I guess it could still world depending on what you are doing.

6625053
Yes that's him!

Twilight and Sunbrust? They are one and the same.

Applejack and Soarin

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What? Straight shipping? Eeeeeww... But if you insist.

Rarity > Prince Blueblood (It can work.)

Fluttershy > Iron Will

Twilight or Pinkie Pie > Time Turner

Applejack > Soarin (This comes up surprisingly often considering their one and only interaction involved his love of apple pie. Wait, nevermind, that's not so surprising.)

Rainbow Dash > Big Mac (I've seen it done a few times.) or Zephyr Breeze (Though I've only seen this one interpreted as a gag ship.)

6625068
Glad I could help.

6625082
Soarin just really gets around doesn't he?

6625094
Same could be said for Mac

6625038
When I realize that ‘common’ and ‘straight ships’ in the same sentence doesn't compute in my brain.

Seriously, though: Most of the Mane Six have been shipped with Soarin and Big Mac, mainly because they're the first eligible male characters to get any screen time. On specific ponies:

Rarity's biggest male ship is Spike. Before you go ‘but he's too young!’, a lot of these are either set in the future, or in alternate timelines where Spike is older when they first meet.

Fluttershy's biggest male ship is Discord, although Big Mac was dominant before Discord's reformation.

I frankly don't see Pinkie shipped that often outside of the Mane Six; I mean, Cheese Sandwich occasionally, but usually that's in the background rather than the central romance of the story.

Dash's biggest male ship is Soarin, by far.

I haven't seen AJ shipped with many male characters; like Pinkie, Caramel is sort of the default used when someone is writing a story focused on a different character and wants to mention AJ is in a relationship.

Twilight, I think, also favors Discord, with a dose of Big Mac. Flash Sentry isn't exactly popular.

This said, see the beginning of my comment for how much I know about straight shipping.

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Fluttershy... Kratus. It's not a common ship, but it ought to be. Look him up; he was only named in the card game.

6625087
Fluttershy and Ironwill is certainly an interesting ship. At the very least if would be entertaining. JUST DO IT!

6625234
I'm not sure who that is. I tried looking it up but all I'm getting is god of war stuff.

6625227
Yes. That’s how it started.

6625252
The salespony from Trade Ya!

You know, the one with the two-headed dog, and the Discord lamp. He's got a common interest with Fluttershy, his means and methods are different enough to make a conflict, his animals might be scarier than what she's used to, but there is, at least, some possible chemistry there.

And the not finding anything is my bad, sorry: apparently you write his name with a 'C'.

6625270
Ok, now I see why I kept getting god of war pop ups. And speaking of Trade Ya, I read one command that said they shipped Fluttershy with Stellar Eclipse.

Really, not one mention of Braeburn so far?He's been shipped with most ponies, even his cousin.

Twilight and Sunburst is a nice ship. Though he is often shipped with Starlight instead.

Rarity's most common ship has to be Spike.

Rainbow Dash and Zephyr Breeze is a bit of a troll ship, much like Rarity and Blueblood and Flashlight. It could be made to work with a more mature Zephyr though.

The kind of love triangle between AJ, Rarity and Trenderhoof got some air.

And last but certainly not least: Applejack and Steelhoves/Applesnack

6625075
Nah, that's more Starlight's straight ship.

6625627
Maybe. But I seen the more as friends. Twilight and sunburst actually look like they could be a couple. Considering that they are both bookworms, very smart and have a lot of other things in common.

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I'd prefer pairing Rainbow up with Thunderlane.

Actually, I've had plans for shipping stories for both Applejack/Caramel and Rainbow/Thunderlane, though both of them involved putting a twist on the stallion to make it more interesting.

Twilight/Flash Sentry is easy, though there are viable alternatives.

Always had trouble shipping Rarity with anyone, to the point where I had to invent a love interest for her while planning one of my stories.

Never been sure how to ship Pinkie and Fluttershy. For some reason I don't like the idea of shipping Pinkie with another crazy party pony like Cheese, but that leaves me without much to choose from. And Fluttershy is the one Mane Six member I've sorta designated headcanon gay - for some reason I can't imagine shipping her with a dude.

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Actually, Stellar would be my second choice for Twilight after Flash. I dunno, he seems kinda adorkable so they'd be a nice fit. Probably into astronomy going by his name and cutie mark. Also owned a huge collection of novelty Discord lamps, which suggest he has some odd interests. So, they'd probably have stuff in common.

6625038 There was a time when RarityxThunderlane was a popular ship, largely because of "Wonderbolts Academy". I also surprisingly saw some stuff involving RarityxBig Mac up to a few years ago. And RarityxFancy Pants seemed to get some attention at one point.

I think Applejack used to get paired up with Caramel a lot, largely because he was the only stallion that really had any kind of connection or on-screen interaction with her.

6625038 Discord is pretty common with Fluttershy. I've seen a few King Sombra ships with Twilight. Also I'm pretty sure Big Mac has been shipped with all of them (which is creepy in Applejack's case).

6626106
How could I forget King Sombra?! Seen him shipped mostly with either Twilight or Fluttershy, though I can't see it for myself.
He'd work great with a slightly corrupt Rarity though.

6626228 Indeed, I think there might be a gew sombrarity shipfics out there.

6626242
Sombrarity sounds like a hat, but I guess that's appropriate for our favourite fashionista.

6626106
Sombra x Twilight is one of my favorite ships, but using it can be difficult without making that the focus of the story.

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What? Straight shipping? Eeeeeww... But if you insist.

I suppose you don't like it on stories, right?

6626427
Nah, I was just playing to a type. I don't actually care.

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6625211
Sorry for ruining the fun but Big Mac already has a canon ship.

Although those other ships were also nice.

6626455
Oh, ok then. No problem.

6626459
Screw canon. They're called Alternate Universes.

6626481
I don't have any problem with that, as I also like Alternate Universes.

6625520
Oh, which reminds me:

Fluttershy and Bulk Biceps. That's a straight ship with some popularity, too. Flutters seems to be paired with the 'cuddly bruiser' type pretty easily.

Fluttershy with either Discord or Big Mac is really popular. Personally I prefer her and Discord

6626459
Not all relationships last. But yeah, depending on when the story is set I guess you may need to take canon into consideration.

6625038
Does Rule 63 shipping count?

Still see sparity around a lot. I've also seen quite a few stories that shipped Twilight and Spike. Obviously stories set some years in the future... at least the ones that aren't... weird.

It's funny though. There aren't all that many male ponies around, and as a result; most the well known ones have multiple ships with members of the main six.

Poor Big Mac has a ton of different ships. Course, considering they might be related, the Big Mac/Marble Pie ship is just a little weird... but I guess it is a distant relation. On a side note, I think it's funny that not many people seem to realize that Pinkie and Marble are twins. Obviously not identical, but twins none the less.