What are the drop rates for the different rarities in Loot Boxes?

Solution 1:

There are no published drop rates as such but here is a link to a Reddit thread which has compiled a list of percentages over 1000 crates by various people compiled into one list. It's still a small sample size in the grand scheme of things but it's better than individual stats. To follow the links below you can use the Reddit thread.

Video link          Common  Rare    Epic    Legendary
Force               235     125     31      9
Royal               113     66      15      6
Futile              116     67      15      6
Jeel                118     64      14      4
Boogie2988          117     64      18      5
DrFeelgood          121     60      13      6
Masier              117     64      14      5
MFPallytime         237     132     27      8
Stylosa             245     125     28      6
Drift0r             116     58      15      7
Redmercy            234     125     33      12
Nero                238     121     29      8
Outfoxed            284     167     31      14
Gamespot            121     65      14      4
Total count         2412    1303    297     100
Percent of Total    58.66%  31.69%  7.22%   2.43%
Drop rate           97.08%  78.22%  25.91%  9.38%

Solution 2:

Blizzard (China) just revealed some numbers, at least for now it is not know if these numbers apply for China only, or for China and any other region.

• Every Loot Box contains one item of Rare (blue) or higher quality 100% of the time.
• For every 5.5 boxes opened (18.2%), a Epic item (purple) will be inside.
• For every 13.5 boxes opened (7.4%), a Legendary (gold) item will be inside.

It is believed that these numbers are valid worldwide but this is not yet confirmed. They are confirmed to apply for China. I'll update as soon as more information is available.

Blizzard did not actually release percentages, but they were calculated by a few places, including PC Gamer.