My favorite right now is regeneration + thick skin. The two abilities complement each other so well that I'm kind of surprised this isn't an official combo.

This is an easy one, but Gracefulness + Fragrance is a pretty crazy combo ^^ What is that, like a 120% dodge chance?

Acute Eyesight + Silent Pace + Saber-toothed results in a +70% crit chance for pumas.

Sharp Feathers + Massive Wings makes Rush awesomely powerful for rocs, both adding damage and a potential knockdown.

Acute Eye + Watchman on scout throdamas for free explores with increased vision (Acute eye gene, Watchman from the morph). Add in Quickmind + Agile paws for increased vermillion magic seal chance. 

Lionsheart + Steel Muscles for really big numbers. +dmg +crit for even bigger numbers.

Seconding Music beak + Massive Wings + Sharp Feather combo. Its fun stuff.

Tireless Heart + Berserker

Tireless heart lets me go forever in the arena and then the berserker is like the sweet cherry for the already delicous ice cream.  

(Sorry, I really want icecream... and I have none :'(   )

Not sure what you mean by official combos, is it like the Breaths or Counterstike, or is it more of what people usually use, but if it's the prior, then:

Light bones + Agile paws -- increases Agility by 50%, who needs defense when you can dodge, right? 

Berserker + Second Breath -- Yup, going again and pumping up my strength. (maybe)

Regeneration + Venomous Skin / Magic Mirror -- It's pretty easy to figure out why :P (Also add in Steel Bones)

Regeneration + Stench / Magic Horn + Gracefulness + Fragance / Muscial Beak + Fragance -- Yeah, going there, dodging and/or stopping plus you heal.

Ditto what Aelisk said --Musical Larynx + Massive Wings is real neat, same with Lionheart + Steel Muscles -- There is also Musical Beak + Second Breath, for Gryphons.

Here are some Official combos + genes

Breath attack + Fearsome Look / Petrifying Gaze -- stops the opponent while dealing massgive damage.

Also Breath attack + Acute Eyesight -- Pretty much 1 hit K'Os if the opponent has only 300 health, most of the time at level 50.

Counterstrike + Gracefulness + Fragance -- Dodging and hitting them back. (Add Acute Eyesight for crits) (One upping mosiac)

Lastly, not sure if this works, as I haven't breed for the genes yet, but:

Gracefullness + Magic Mirror -- Don't know if dodging lets you also reflect it back or not, but if it does, then this is pretty much a weaker version of Counterstrike, with only 2 genes, instead of 3.

Edit: Reading about Thordama morphs got me searching for what they do, and darn, I missed out on quite a few potential combos (Wish I could see Morphs in the Gene List.) Anyways some are:

Scout Morph + Acute Eyesight -- +60% critical, nearly the same as a Puma's

Harvester + Steel Muscle -- It's like you have two Steel Muscle genes now!

Ps. While writing my combos down, realized, once more, how strong a complete genetic Unicorn can be.

THERE IS SO MUCH TO LEARN