What rules determine the apostrophe placement in "ham 'n eggs" and similar expressions?

In expressions such as "ham 'n eggs", the conjunction 'n appears to replace and, yet there is only one apostrophe to indicate the missing a and none for the missing d (i.e., no "ham 'n' eggs").

Is there a punctuation rule that governs this or is it just an idiosyncratic case?

From a cursory glance, it appears to be a rule up to free variation. Consider:

1. From a naive search with the Google N-Gram viewer, it appears that n' has a slight edge.

But if you zoom in to look at variations of and eggs, as well as ham and eggs, the picture gets murkier. Keep in mind that, since this is very informal usage, that it probably wasn't used frequently in the print media that the Google Ngram viewer searches through.

2. Examples from popular culture:
   • Linens 'n Things
   • Guns N' Roses
   • Steak 'n Shake
   • "Hot n Cold"
3. Variations of Ham and Eggs, culled from search engines:
   • Ham N' Egg restaurant
   • Ham N' Eggs restaurant
   • Jackie's Ham 'n Eggs restaurant
   • "Ham 'n' Eggs" rap song
   • Ham 'n Eggs and Ham 'n Eggers referenced by the San Diego Herald, referring to the California pension initiative in 1938-1939

Conclusion: People tend to do one of 4 things to denote an unemphasized and:

1. 'n'
2. 'n
3. n'
4. n or N

As there is no broad consensus, and because people don't think about edge cases like these, I believe you could justifiably use any of them.

If there'd've been a rule against multiple apostrophe I'd've known about it. Or somebody'd've.

More here.