Using apostrophes for contractions should not be done in formal writing.  Therefore, these rules for apostrophes cover possession.

Rule 1: Singular Possession
To show possessions for singular nouns that do not end in -s, use an -'s.
The dog's bone    The man's job   Mary's cane

To show possession for singular nouns that end in -s, use -'s or just -' but be consistent.
Chris's books   or   Chris' books

Rule 2: Plural Possession
To show possession for plural words that end in -s, use only an apostrophe.
 Two dogs' bones    The guys' night out    The ladies' apartment
 
To show possession for plural words that do not end in -s, use an -'s.
The children's dreams      The people's champion

Rule 3: Joint Possession
To show joint possession (both nouns own the same thing), use -'s with only the last noun.
 
Example: Jen and Mike's kids will be very tall.
 
Example: Staci's and John's modes of transportation are very different; Staci takes the train while John drives a new car.
 
In the first case, the two people share the children. In the second case, each person has his or her own method of travel, so it is not joint possession.
 
Exercise
 
1. Paul's and Susan's dog got out of the yard.

Paul and Susan Paul and Susan's correct

2. The Johnson's moved across the street.

Johnsons' Johnsons correct

3. The cars' tires were stacked near its rear bumper.

car's cars correct

4. Jim and Ryan's pets houses were next door to one another.

Jim's and Ryan's Jim and Ryan correct
 
Answers
1.  Paul and Susan's (joint possession; they own they dog)
2.  Johnsons (no possession exists)
3.  car's (one car as indicated by the "its")
4.  Jim's and Ryan's (each owns his own home)