This is easy for me, given where I grew up and what you read in the intro:

1. Whitaker/Trammell
2. Morgan/Concepcion
3. Alomar/Vizquel

As you nominate or vote for combos, try to make sure they played together for at least one full season.

1. Aparacio/Fox
2. Alomar/Vizquel
3. Alomar/Fernandez

1. Morgan/Concepcion
2. Whitaker/Trammell

(big drop-off)

3.   Alomar/Vizquel

The #3 selection is a dartboard choice.  Gordon/Rizzuto and Doerr/Stephens would be right up there, and I'd have to think very, very hard about defence.

I don't think the Alomar-Vizquel tandem has ever received the proper respect. They weren't together all THAT long I suppose (which would have an effect on their 'feel' for each other). But you'd have to argue with me for a looong time to convince me that those two didn't make up the best middle infield duo in the history of the game.

I will encore Vizquel - Alomar as the best in recent history.

Since the question is phrased the way it is,
I couldn't vote against Trammell/Whitaker

Hmm... tougher than the pitchers as there are many second basemen and shortstops who were fantastic but rarely are two greats playing together.

Jay keystones worth thinking about were Damiso Garcia/Alfredo Griffen, Garcia/Tony Fernandez, Roberto Alomar/Manny Lee, Alomar/Alex Gonzalez, Homer Bush/Gonzalez, Orlando Hudson/Chris Woodward, Hudson/Russ Adams.  Alomar/Fernandez would've been the best but Fernandez was here just 1/2 a season while Alomar was (they were traded for each other).  Otherwise I'd go with Garcia/Fernandez as both were All Stars while on the team together.

So I think we can safely remove the Jays from consideration. 

Great shortstops...
Cal Ripken/Roberto Alomar - together for 96-98.  130+ OPS+ for Alomar in 2 of those seasons, Cal above 100 just once.
Ozzie Smith/Tom Herr - solid combo but not earth shattering
Barry Larkin/Brett Boone?  Never seemed to play with anyone great
Ernie Banks/?  played with a lot of also rans

Hrm.  Not coming up with anything yet. 

My vote will go to Whittaker/Trammell until I find a better combo.

Using the ever handy Baseball-Reference site, here are some other noteworthy combos:

Hornsby/T. Jackson

F. Frisch/T. Jackson

Joe Gorden/Lou Boudreau

 Jackie Robinson/Pee Wee Reese

Peter North and Herschel Savage.

Oh...wait. Oops, my bad. Should have read more than the headline.

Ooh, Robinson and Reese! Good one ... my mom, the Brooklyn Dodgers fan, would be disappointed I didn't think of that one myself!

I think Trammell/Whitaker, even with neither in the HOF -- one should be, the other could be -- still just noodges them out, but it's thisclose.

I couldn't find anyone of note to pair with Rod Carew while he played 2B. I don't think Danny Thompson will make any all-time greats list.

Mike Bordick/Orlando Hudson

I'll put in my vote now that I've seen everyone else's:

Trammel/Whitaker
Reese/Robinson
Morgan/Conception

I hereby revise my second vote to be for Reese/Robinson, and bump everyone else down a spot (so Alomar/Fernandex get bumped)