It's less WTF-inducing than Reign and the designer seem to understand the basic elements of 16th century clothing, but a lot of liberty have been taken with the costumes. Keep in mind though this is not a documentary, the plot and the storyline already deviated far from history so I don't think it's fair to criticize the inaccuracies in costume designs. It's in the same realm as other "historical fantasy" series. Many fashion history enthusiasts can go on and on about why pretty much everything is wrong. I'm not that nitpicky but I can point out some of the more outrageous headwears that I think everyone can laugh at:
Henry wore a very familiar looking crown, it can be yours for $3 at Halloween section.
Anne Boleyn is so edgy, she folded a napkin and wore that as a hat
A group of Soviet spies tried to infiltrate the court, but their cover was blown when Henry noticed they wore these Russian headwear thingys instead of French hoods.
Outraged at her potential loss at the hat game, Mary traveled to the 1910's and stole the hat from Downton Abbey's Dowger Countess.
And then there's poor Anne of Cleves. To be fair historians did say that she did not receive formal education, couldn't speak English well, and was really unfamiliar with English culture. We didn't expect Queen Amidala's wardrobe (she even has an antenna hat).
Bonus: Victoria's Secret show, 1520
Not very. One of the big problems with costuming in The Tudors is that a lot of the clothing items that are relatively accurate are from a slightly later period, about fifty years on. Most likely this is because the Elizabethan era is more popular for films and plays, and therefore the costume closets have a lot more of that clothing than proper Henrician outfits. This outfit on Katherine of Aragon, for example, is from the 1550s. This ruff on Anne Boleyn and this one on Jane Seymour are also several decades early. Padded shoulder rolls, basques, shoulder puffs, long fitted sleeves ... these are all anachronistic for the 1530s. What women should be wearing are gowns with long sleeves that are fitted at the shoulder and flare out over decorated undersleeves, plain bodices (no faux stomachers or weird embroideries), their hair parted down the middle, pulled back, and hidden. Men's clothing should emphasize broadness and bulk, and there should be codpieces.
Here is the FrockFlicks take on common problems with 16th century costuming, much of which is applicable to The Tudors.