In Hearts, played by experts, what is ratio of shooting the moon?
Hearts involves the concept of (A) deciding exactly 3 cards (at least in the common 4 player variant) to change between each hand in 3/4 of the hands and (B) to keep all cards in 1/4 of them. 
A major enough distinction that I think they probably need to considered in these 2 different groups. Lets consider what hand distributions capable of taking all cards would look like and consider group B as more important as it is easier to site and is based off of which players will decide which cards to pass from their hands anyway.

There are a total of 39 distinct hand suit patterns, there is a large scale experiment of computer shuffled hands and results here
http://playbridge.com/pb_shuffle_project.php

These are suit independent so multiplication to increase these percentages is not needed, but certain constraints need to be taken into consideration and will reduce the percentages a significant amount. 
Basically to get the total you would need to add up the favorable results you get from each of the 39 hand patterns to get the total % of being able to shoot the moon.

Below this point might be considered a tedious explanation on how these numbers are derived and in excess of what the question asked for, but in case of more curious readers, I thought I would include the general process as it can be used of any of the 39 hand patterns :) Edit: section added after to account for hand passing as requested.

Lets illustrate with having 2 sample hand patterns for a key in what to look for, will do 5-4-3-1 and 7-3-3-0 as examples.


5-4-3-1, about 12.93% likely to get without caring which cards are in the suits. There are 24 possible arrangements between the suits in this chosen hand pattern,  lets add what restrictions we need 1 suit at a time.


For the 1 card suit we have 13 combinations of those, in 6 of the 24 suit arrangements this suit is clubs and does not need any restriction as a point card cannot be played on the first trick, for the other arrangements having the ace is safest and so would need to divide by 13.
For the 3 card suit we have 13 choose 3 combinations of those (or 286) and only 1 of them realiably has us winning all tricks (akq) and so would need to divide by 286 for all other arrangements.
For the 4 card suit we have 13 choose 4 combinations (or 715) and akqj would need needed a high percentage of the time so will need to divide by 715
For the 5 card suit we can finally have a little leeway; 13 choose 5 combinations (or 1287) and the suit could reliably run with just akqj, having those 4 cards accounted for 
we have 9 choose 1 combinations left so basically divide by 1287 and multiply by 9.

This leaves the result looking visually like this:
5-4-3-1: 12.93% 
A. 3.233% of which has 1 club.
B.  9.967% other arrangements

A. 3.233%/286/715/1287*9 or    1.11e-7 % or 11.1e-8 %

B. 9.967%/13/286/715/1287*9 or 2.55e-8 %

for a total of 1.365e-7 %


7-3-3-0, about 0.2652% likely without caring about which suit or which cards. There are only 12 possible suit arrangements 6 of which have three clubs for instance, again going 1 suit at a time.


For the suit with no cards in it, no restrictions as nothing can be done about it anyway

For the suits with 3 cards 13 choose 3 (287) if one of them is clubs, (edit: this also holds true if the hand has no clubs at all) that suit can have ak and any of the 11 cards, the second 3 card suit for safety should probably be akq; thus the need to divide by 287 twice.

For the 7 card suit, we have 13 choose 7 (1716) combinations, since the suit is substantially longer, we probably only need akq for a relaible run most of the time, akqj for a nearly everytime running leaving 10 choose 4 (210) or 9 choose 3 (84) open slots.

So In total

0.2652% /287/287/1716*210 or 3.94e-7 %



Hand passing section:

For the cases where 3 cards are being passed, people try to shorten their already short suits and lengthen their strong ones. It can be helpful to pair 
your hand with that of the person passing to you, combining their suit lengths with yours, below is a table of the 44 fit classes breifly mentioned in the comment, the first 10 are by far the most common and in cases of passing across cannot be changed by the passing, placing a logical upper bound on how uneven the suits can be distributed.

(from Thomas Andrews Deal 3.1 documentation)
There are 103 fit patterns, or 65 if we consider our fit pattern and opponent's as the same (e.g., that 8-6-6-6 is the same as 7-7-7-5. If the sum of the longest fit and the shortest fit is 13, then the pattern is self-dual, for example, if our fit pattern is 9-7-6-4, then so is the opponent's.)

Normalized |  Squares sum | Patterns

     0 |          170 | 7-7-6-6
     1 |          172 | 8-6-6-6,7-7-7-5
     2 |          174 | 8-7-6-5
     4 |          178 | 9-6-6-5,8-8-5-5,8-7-7-4
     5 |          180 | 9-7-5-5,8-8-6-4
     6 |          182 | 9-7-6-4
     8 |          186 | 10-6-5-5,9-8-5-4,8-8-7-3
     9 |          188 | 10-6-6-4,9-7-7-3
    10 |          190 | 10-7-5-4,9-8-6-3
    12 |          194 | 10-7-6-3,9-9-4-4
    13 |          196 | 11-5-5-5,10-8-4-4,9-9-5-3,8-8-8-2
    14 |          198 | 11-6-5-4,10-8-5-3,9-8-7-2
    16 |          202 | 11-7-4-4,11-6-6-3,10-7-7-2,9-9-6-2
    17 |          204 | 11-7-5-3,10-8-6-2
    18 |          206 | 10-9-4-3
    20 |          210 | 12-5-5-4,11-8-4-3,11-7-6-2,10-9-5-2,9-8-8-1
    21 |          212 | 12-6-4-4,9-9-7-1
    22 |          214 | 12-6-5-3,11-8-5-2,10-8-7-1
    24 |          218 | 12-7-4-3,10-10-3-3,10-9-6-1
    25 |          220 | 12-6-6-2,11-9-3-3,11-7-7-1,10-10-4-2
    26 |          222 | 12-7-5-2,11-9-4-2,11-8-6-1
    28 |          226 | 13-5-4-4,12-8-3-3,10-10-5-1,9-9-8-0
    29 |          228 | 13-5-5-3,12-8-4-2,11-9-5-1,10-8-8-0
    30 |          230 | 13-6-4-3,12-7-6-1,10-9-7-0
    32 |          234 | 13-6-5-2,12-8-5-1,11-10-3-2,11-8-7-0
    33 |          236 | 13-7-3-3,10-10-6-0
    34 |          238 | 13-7-4-2,12-9-3-2,11-10-4-1,11-9-6-0
    36 |          242 | 13-6-6-1,12-9-4-1,12-7-7-0
    37 |          244 | 13-7-5-1,12-8-6-0
    38 |          246 | 13-8-3-2,11-10-5-0
    40 |          250 | 13-8-4-1,12-9-5-0,11-11-2-2
    41 |          252 | 12-10-2-2,11-11-3-1
    42 |          254 | 13-7-6-0,12-10-3-1
    44 |          258 | 13-9-2-2,13-8-5-0,11-11-4-0
    45 |          260 | 13-9-3-1,12-10-4-0
    48 |          266 | 13-9-4-0
    50 |          270 | 12-11-2-1
    52 |          274 | 13-10-2-1,12-11-3-0
    54 |          278 | 13-10-3-0
    60 |          290 | 12-12-1-1
    61 |          292 | 13-11-1-1,12-12-2-0
    62 |          294 | 13-11-2-0
    72 |          314 | 13-12-1-0
    84 |          338 | 13-13-0-0