Everyone loves a beach destination especially one that offers lush, tropical vegetation and pristine lagoons in combination with overwater bungalows and almost near seclusion. Since award availability is opening up for Tahiti's dry season (and high season) next year, which is from April to October, I thought it best to cover your options to get there using miles.
American AAdvantage Miles
You can fly partner airline Air Tahiti Nui using 62.5k AA miles for a one-way in business or 37.5k one-way in economy. Searching for availability on Expert Flyer consistently shows at least 2 business class seats and 6 economy seats available. Best results occur when booking your award as far out as possible. You have to call AA to book this award.
62.5k AA miles also gets you a business class seat on Hawaiian. These are pretty similar to their domestic first product so might not be "aspirational" enough for some. A one-way economy class award costs 37.5k miles. These are bookable on aa.com.
You can use Skymiles to book your award with Air France or Air Tahiti Nui. Expert Flyer shows availability with Air Tahiti Nui 330 days out though delta.com is showing none. Call a Delta rep to check on that and if necessary, convince them to do a long sell for the dates that should have availability. Delta unfortunately charges fuel surcharges on this award.
Air France flies LAX-PPT too. Economy award availability was easy to find though finding more than 1 business class seat can be a mission. Try to book as far out as possible, be flexible, and check back to see if there's space open.
Roundtrip business class costs 150,000 miles, while economy is 100,000 miles.
Alaska Airlines Mileage Plan
Your Alaska Airlines miles can also be used to book Air France awards. Round-trip in Business costs 120,000 round-trip. Call Alaska Airlines reservations to book.
United MileagePlus and US Airways miles
There are no direct flights from the US to Tahiti using Star Alliance carriers. Routing through Auckland and continuing onto PPT via Air New Zealand is your best option. 35,000 United miles are required for economy one-way while 60,000 miles gets you one-way in biz. US Airways requires 80,000 in economy and 110,000 miles in business round-trip. I couldn't find much business class availability though economy was quite abundant.
How to get to the other islands?
Fly the domestic airline Air Tahiti (no association with Air Tahiti Nui) or take a ferry. The ferry from Papeete to Moorea costs $15 and takes just 30 minutes whereas the flight costs ~$60 and only takes 10 minutes. There are also cargo ships to the other islands. Vaeanu and Hawaiki Nui are two that have a passenger cabin but may not be the most comfortable nor reliable. If you plan on visiting multiple islands, the Air Tahiti Pass may be worth looking into.
Mighty's Take: Finding business class seats can be difficult even 330 days out. Best bet would be to use your miles for travel on Air Tahiti Nui. It's the most direct route and you have the option of burning either American AAdvantage miles or Delta Skymiles. Delta does impose fuel surcharges while American does not. Given the difficulty of burning Skymiles, I'd probably burn those first.