I don't recommend using appearance alone as your egg safety test. Particularly with respect to an egg that has been cracked open, there can be a good number of discolorations in the white or yolk that do not represent food safety concerns. However, with that context in mind, there are certain types of egg inspection techniques that make sense when trying to determine whether an egg has gone bad.
First, I would look to ensure that there are no cracks in the shell. If there are cracks, I would dispose of the egg.
Once you have cracked the shell, smell the egg and see if it smells fine. If not, I would throw it away. If the raw white of the egg has turned pinkish, green, or black, you should also definitely toss out the egg.
In terms of determining quality from an uncracked egg, in his classic book, On Food and Cooking (Scribners, 2004, Revised Updated Edition) Harold McGee discusses how better quality eggs have smaller air cells; he notes that in a fresh egg, the space is about 1/8-inch deep and has the diameter of a dime. To test the size without cracking it, he writes that you should put the whole egg in a bowl of water; if the air cell is much larger than the size noted above, the wide end of the egg will rise above the narrow end. If your egg does this, it would then be a reflection that it is not that fresh.
McGee's freshness test is also related to a long-time folklore belief about the degree to which an egg will float in water. According to this belief, a spoiled egg will float all the way up to the surface but a safe egg will not.
While this belief is not completely accurate, it does harbor one important element of truth. If the pores in an eggshell have become sufficiently large, or if the shell has been cracked, it is possible not only for air but also for additional bacteria to enter into the egg, become metabolically active, and create gas inside the eggshell. This additional gas could cause the egg to be more buoyant. (However, other factors could also cause the egg to float, including simple transfer of air through larger pores, even if no bacteria had migrated into the egg.)
The bottom line: you may want to try the float test for fun or the "wide end up" float test for freshness. But to determine an egg's safety, judge by the smell and discoloration of the opened egg as described above and notice if there are cracks in the shell of the unopened egg.
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