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MySQL, varias filas para separar campos

Solo te falta un grupo por :)

SELECT PartNumber,
  MAX(IF (Priority = 0, SupName, NULL)) AS Sup1,
  MAX(IF (Priority = 1, SupName, NULL)) AS Sup2,
  MAX(IF (Priority = 2, SupName, NULL)) AS Sup3
FROM SupXref
GROUP BY PartNumber

Editar:

Después de jugar un rato, creo que obtuve la primera solución que estabas buscando. Pruébalo :)

SELECT partnumber,
  COALESCE(Sup1, COALESCE(Sup2, Sup3)) AS Supp1,
  IF (Sup1 IS NULL, IF (Sup2 IS NULL, NULL, Sup3), COALESCE(Sup2, Sup3)) AS Supp2,
  IF (Sup1 IS NULL, NULL, IF (Sup2 IS NULL, NULL, Sup3)) AS Supp3
FROM (
  SELECT PartNumber,
    MAX(IF (Priority = 0, SupName, NULL)) AS Sup1,
    MAX(IF (Priority = 1, SupName, NULL)) AS Sup2,
    MAX(IF (Priority = 2, SupName, NULL)) AS Sup3
  FROM SupXref
  GROUP BY PartNumber
) AS S

Para la siguiente tabla:

+------------+----------+---------+
| PARTNUMBER | PRIORITY | SUPNAME |
+------------+----------+---------+
| a1         |        2 | Three   |
| a2         |        1 | Two     |
| a3         |        2 | Three   |
| a3         |        1 | Two     |
| a4         |        0 | One     |
| a5         |        0 | One     |
| a5         |        2 | Three   |
| a6         |        0 | One     |
| a6         |        1 | Two     |
| a7         |        0 | One     |
| a7         |        1 | Two     |
| a7         |        2 | Three   |
+------------+----------+---------+

Los datos se convierten en esto:

+------------+------+------+-------+
| PARTNUMBER | SUP1 | SUP2 | SUP3  |
+------------+------+------+-------+
| a1         |      |      | Three |
| a2         |      | Two  |       |
| a3         |      | Two  | Three |
| a4         | One  |      |       |
| a5         | One  |      | Three |
| a6         | One  | Two  |       |
| a7         | One  | Two  | Three |
+------------+------+------+-------+

Y finalmente en esto:

+------------+-------+-------+-------+
| PARTNUMBER | SUPP1 | SUPP2 | SUPP3 |
+------------+-------+-------+-------+
| a1         | Three |       |       |
| a2         | Two   |       |       |
| a3         | Two   | Three |       |
| a4         | One   |       |       |
| a5         | One   | Three |       |
| a6         | One   | Two   |       |
| a7         | One   | Two   | Three |
+------------+-------+-------+-------+