This is the sort of challenge that bytes

Jelly, 8 programs

Ṣ                   Built-in sort.
¹Þ                  Sort-by the identity function.
Ụị                  Sort indices by values, then index into list.
Œ!Ṃ                 Smallest permutation.
7778Ọv              Eval Unicode 7778 (Ṣ).
ẊI>@-.S$$¿          Bogosort.
<;0œṡ0⁸ṁjµ/         Insertion sort.
AṀ‘¶+Ç©ṬT_©®³ċЀ®x' A terrifying hack.

The last program is really annoying…

AṀ‘¶+Ç©               Add ® = abs(max(L)) + 1 to the entire list.
                      Now it’s offset to be entirely positive.
       Ṭ              Create a binary array with 1s at these indices.
        T             Find the indices of 1s in this array.
                      The result is sorted, but offset wrong, and lacks duplicates.
         _©®          Subtract the offset, saving this list to ®.
                      Now restore the duplicates:
            ³ċЀ      Count occurences in the original list.
                ®x'   Repeat the elements of ® that many times.

If I can remove the œṡ from <;0œṡ0⁸ṁjµ/, there’s also this weird one: ²SNr²ZFœ&. Help is appreciated.


Jelly, 10 programs, 65 bytes

Ṣ
¹Þ
Ụị
Œ!Ṃ
7778Ọv
Ẋ>2\S$¿
ĠFḣṪ¥@€
~Ṁ~rṀxLœ&
C»/ð+ÆNPÆfÆC_ḷ
<þḅ1‘WiþJḄ³ṫZḢ

There's some inevitable overlap with @Lynn's Jelly answer. Credits for the bogosort idea go to her.

Try it online! or verify uniqueness.

How they work

Ṣ               Main link. Argument: A (array)

Ṣ               Sort A.
¹Þ              Main link. Argument: A (array)

¹Þ              Sort A, using the identity function as the key.
Ụị              Main link. Argument: A (array)

Ụ               Grade up; yield all indices of A, sorted by their corr. values.
 ị              Index into A.
Œ!Ṃ             Main link. Argument: A (array)

Œ!              Yield all permutations of A.
  Ṃ             Minimum; yield the lexicographically smallest permutation.
7778Ọv          Main link. Argument: A (array)

7778Ọ           Unordinal; yield chr(7778) = 'Ṣ'.
     v          Evaluate with argument A.
Ẋ>2\S$¿         Main link. Argument: A (array)

      ¿         While the condition it truthy, execute the body.
 >2\S$            Condition:
     $              Combine the two links to the left into a monadic chain.
 >2\                  Perform pairwise greater-than comparison.
    S                 Sum; add the results.
                    This returns 0 iff A contains an unsorted pair of integers.
Ẋ                 Body: Shuffle A.
ĠFḣṪ¥@€         Main link. Argument: A (array)

Ġ               Group the indices of A by their sorted values.
 F              Flatten the result.
      €         Apply the link to the left to each index in the previous result, 
                calling it with the index as left argument and A as the right one.
    ¥@            Combine the two links to the left into a dyadic chain and swap
                  its arguments, so A is left one and the index i is the right one.
  ḣ               Head; take the first i elements of A.
   Ṫ              Tail; yield the last of the first i, i.e., the i-th element of A.
~Ṁ~rṀxLœ&       Main link. Argument: A (array)

~               Take the bitwise NOT of all integers in A.
 Ṁ              Take the maximum.
  ~             Take the bitwise NOT of the maximum, yielding the minimum of A.
    Ṁ           Yield the maximum of A.
   r            Range; yield [min(A), ... max(A)].
      L         Yield the length of A.
     x          Repeat each integer in the range len(A) times.
       œ&       Take the multiset-intersection of the result and A.
C»/ð+ÆNPÆfÆC_ḷ  Main link. Argument: A (array)

C               Complement; map (x -> 1-x) over A.
 »/             Reduce by dyadic maximum, yielding 1-min(A).
   ð            Begin a new, dyadic chain. Arguments: 1-min(A), A
    +           Add 1-min(A) to all elements of A, making them strictly positive.
     ÆN         For each element n of the result, yield the n-th prime number.
       P        Take the product.
        Æf      Factorize the product into prime numbers, with repetition.
          ÆC    Prime count; count the number of primes less than or equal to p,
                for each prime p in the resulting factorization.
             ḷ  Yield the left argument, 1-min(A).
            _   Subtract 1-min(A) from each prime count in the result to the left.
<þḅ1‘WiþJḄ³ṫZḢ  Main link. Argument: A (array)

<þ              Construct the less-than matrix of all pairs of elements in A.
  ḅ1            Convert each row from base 1 to integer (sum each).
    ‘           Increment. The integer at index i now counts how many elements
                of A are less than or equal to the i-th.
     W          Wrap the resulting 1D array into an array.
        J       Yield the indices of A, i.e., [1, ..., len(A)].
      iþ        Construct the index table; for each array T in the singleton array
                to the left and index j to the right, find the index of j in T.
                This yields an array of singleton arrays.
         Ḅ      Unbinary; convert each singleton from base 2 to integer, mapping
                ([x]-> x) over the array.
          ³     Yield A.
           ṫ    Tail; for each integer i in the result of `Ḅ`, create a copy of A
                without its first i-1 elements.
            Z   Zip/transpose. The first column becomes the first row.
             Ḣ  Head; yield the first row.

05AB1E, score = 6

05AB1E uses CP-1252 encoding.

Thanks to Kevin Cruijssen for program 4.
Thanks to Riley for inspiration to program 6.

Program 1

{               # sort          

Try it online!

Program 2

`[Ž.^]¯

`               # flatten input list to stack
 [Ž  ]          # loop until stack is empty
   .^           # add top of stack to global list in sorted order
      ¯         # push global list

Try it online!

Program 3

WrZŠŠŸvy†

Wr              # get minimum value in input list and reverse stack
  ZŠ            # get maximum value in input list and swap move it to bottom of stack
    Š           # move input down 2 levels of the stack
     Ÿ          # create a range from max to min
      v         # for each y in range
       y†       # move any occurrence of it in the input list to the front

Try it online!

Program 4

ϧ

œ               # get a list of all permutations of input
 ß              # pop the smallest

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Program 5

êDgFDNè¹sUXQOFXs}}\)

ê                      # sort with duplicates removed
 Dg                    # duplicate and get length
   F                   # for N in [0 ... len-1] do
    DNè                # duplicate and get the Nth element in the unique list
       ¹s              # push input and move the Nth element to top of stack
         UX            # save it in X
           Q           # compare each element in the list against the Nth unique element
            O          # sum
             FXs}      # that many times, push X and swap it down 1 level on stack
                 }     # end outer loop
                  \    # remove the left over list of unique items
                   )   # wrap stack in a list

Try it online!

Program 6

©€Ý逤þ((®€Ý逤(þ(Rì

©                        # store a copy of input in register
 €Ý                      # map range[0 ... n] over list
   é                     # sort by length
    €¤                   # map get_last_element over list
      þ((                # keep only non-negative numbers
                         # now we have all positive numbers sorted
         ®€Ý逤(þ(       # do the same thing again on input 
                         # except now we only keep negative numbers
                  R      # reverse sorting for negative numbers
                   ì     # prepend the sorted negative numbers to the positive ones

Try it online!