Holy PvE / Holy PvP
This is one of the dual spec options where a player won't necessarily be switching roles, but rather choosing talents that are optimized for one aspect of the game but not the other. The typical Holy raiding build is a 53/ 0/ 18 complementing the Holy tree with Retribution all the way to Sanctified Seals for higher crits with healing spells, a healing Paladin's bread and butter.
PvP builds vary much more than raiding builds, with one kind of build going 53/ 18/ 0 up Protection instead of Retribution for Improved Hammer of Justice. Others put two more points in Protection 51/ 20/ 0 for Divine Guardian. Still popular builds forego the 30 second stuns and Beacon of Light for Divine Purpose in a 49/ 0/ 22 build that's closer to the raiding specs.
Protection PvE / Holy PvE
This is likely one of the more popular raiding dual spec combinations, particularly for tanks. The typical raiding build for Tankadins is 5/ 60/ 6 with points in Holy for Seals of the Pure and Retribution for Deflection. The Protection tree looks to change somewhat in Patch 3.1, hopefully to lessen the bloat. While Paladin main tanks are unlikely to pick up these options, off- or secondary tanks should find it useful to be able to contribute on the healing end when their tanking skills aren't needed for the fight. In many cases, an extra healer is more welcome than another DPS.
Protection PvE / Retribution PvE
Still, a few tanks would welcome the chance to put away the shield for some fights and swing around a big 2-hander instead. This should also be a popular dual spec combo for raiding for both off-tanks and Retadins, who will sometimes be called upon to tank something. The typical Retribution raiding build is a 0/ 10/ 61 with the bare minimums in Protection -- Divine Strength and Blessing of Kings. This should free up some talent points when Blessing of Kings becomes baseline in the same patch.
Retribution PvE / Holy PvE
On the other hand, some raiding Retadins would love to lend a healing hand in certain fights. Or vice versa, during fights that are don't need much healing, or for DPS races, some healers might want to switch to a DPS role to help speed things up. Depending on what the raid needs at the time, DPS / Healer hybrids can toggle between specs to fill a certain role.
Retribution PvE / Retribution PvP
Personally, this is where I've been blowing my Gold on. I had sometimes been speccing to heal, but ever since I picked up the best melee weapon available it just didn't make sense to the guild for me to raid as Holy. This cut my costs down by a third, but it's still expensive. Retribution PvP spec loses a lot by going deeper into Protection for Improved Hammer of Justice and Divine Guardian (for Arenas only. You can stop at 18 points for solo PvP) in a 0/ 20/ 51 build. Builds vary, with about 12 discretionary points placed in different talents depending on your play style. PvP spec loses out on raid utility and about 12-14% damage by losing Righteous Vengeance.
Protection PvE / Protection PvP
I'm going out on a limb here by predicting one offbeat dual spec combination -- Protection in two flavors. If the announced changes to Paladins in Patch 3.1 push through, we'll have very frightening Protadins who are hard to kill and stun you all the freaking time. This requires two specs, though, because the ideal tanking build will be skipping the PvP talents Stoicism, Guardian's Favor, and Improved Hammer of Justice. No build for Protection PvP just yet, we'll have to wait for 3.1 for that.
Protection PvE / Retribution PvP
This was a fairly common choice of two specs, as many tanks would unwind as Retribution in the Battlegrounds. I suspect that this dual spec combo won't be as popular come Patch 3.1 because of the sudden viability of Protection in PvP. Considering that tanks will have the best shields and one-handers available to them, why bother with the vanilla Crusader Strike and Divine Storm when they have Shield of Righteousness, Hammer of the Righteous, and an Avenger's Shield that can silence?
Holy PvE / Retribution PvP
Let's face it, Retribution is the face-smashing spec. Many players still consider Retribution as the PvP spec, so I know a lot of healers who spec Ret in order to PvP. This isn't absolutely necessary anymore considering Holy Paladins have far greater survivability than Retribution in Arenas and in general. They also have potent offense now, and look to have even better damage capabilities with buffs to talents in Patch 3.1.
Other dual spec combos
There will be some oddball combinations, like raiding as a Retadin and PvPing as a Protadin, or strange cases like Holy PvP and Retribution PvP for filling out roles in Arena teams and solo play. There will even be dual spec combos for raiding and farming -- some Holy Paladins would perhaps like to AoE grind during raid downtimes. When dual specs kick in, a world of options open up for the class, allowing players to choose two of their favorite playing styles for the price of one.
Because glyphs are part of the equation, this again saves a lot of money -- although Inscribers (it's a real word, so please no more comments about that) will likely feel the crunch as players will purchase glyphs once. PvE and PvP glyphs are different as are the optimal glyphs for each spec. It should be interesting to see which glyphs will rise to prominence, or if we'll see new glyphs in Patch 3.1 that will cater to new play styles. Overall, Patch 3.1 will be a turning point in the game for Paladin players. Many will be encouraged to expand their horizons, even trying out specs they've never tried before. As useful as Paladins are now, they'll get even more utility when dual specs finally arrive. Good times.