This is for you hard-core Scabble players who know the rules, not clowns like me who play once in a while.
If you add a letter to the end of a word already on the board can you also put down a word at right angle to the existing word? For example, “play” is on the board and you add an “s” to make it “plays”; can you at the same time put “afe” after the “s” so it looks like:
p
l
a
y
safe
Yes. As long as you’re only going in a straight line at all times. (You couldn’t add an “R” under the “e,” for example.)
Absolutely. In fact that’s a good reason to hold on to your esses–it increases the amount you get on one play significantly.
Any extraness of the spot the ess goes on (e.g., double letter or double word) gets doubled, too, since it’s read both ways.
And of course you can add a word with an ess somewhere in it to the bottom of “play”; it doesn’t have to only start with ess. If that makes sense!
Yep, leaving plurals available in this way can be a mistake if the line it opens up is a high scoring one.
In effect, what you are doing is putting a new word on the board, in such a fashion that one of its letters extends an existing word; in which case, both words score. However, bonus squares count only if they have been played on in that turn. In your instance, if “S” was on a triple-letter score then it would count as a tripled letter in both words; if a double-word score, then both words would score double.
Similarly, you can put a word down parallel to (and touching) an existing word, as long as all of the two-letter words so made are legitimate. Suppose the word “karma” is already on the board and you put down “entreat”, like this:
karma
entreat
This is legal because it makes the words “me” and “an”, and you score for both of these, plus the word “entreat”, plus the 50-point bonus for using all your letters. Again, any special squares under the “e” or the “n” count twice.
You could also build off the “PLAY” with the “S” by making “SPLAY.” 
Have you seen Super Scrabble? Twice as many letters, a bigger board. You can now make QUIZZICAL on a quadruple-word score, use up all your letters, and take the fifty-point bonus.
This is my dream.
But in this example, if there’s a double-word square under f, does the double word apply to both words?
(My SO’s family rules says yes, I think it applies to “safe” but not to “plays”.)
I’d assume just to “safe” because “plays” has no relation to the double-word square.
No, it absolutely does not. Bonuses only count the first time a letter is placed upon them, for words that that letter is a part of. Your interpretation is correct.
But that’s not why it doesn’t count. The person is playing safe as a double word and it is indeed the first time they played it. The question is should the same scoring apply to the second word (“Plays”).
No. This is quite clear in the official Scrabble rules.
Well, they can play with any house rules they choose, but if they think those are the official rules, they’re wrong. Your analysis is correct.
You overlooked the second part of his answer, which says [only] for words that that letter is a part of. That letter (which is on the double-word square), in this case the “f,” is not part of “plays.”
To put it more simply, the answer is no, because none of the letters in the word “plays” is on that double-word square.
<hijack>I think this is a good time to add my theory that playing Scrabble TM and doing crossword puzzles from an early age really does make for a more intelligent adult.</hijack>
No offense Dan–but did you ever consider, say, looking at the box? It should have all the rules printed on the inside. Legal plays and how to score them is pretty basic and covered quite well.
to remember how this rule works , think of it this way:
if you put a tile on the colored square, and that tile is part of 2 words, then the double points apply to both words
But the colored squares (i.e. double word, triple letter, etc) count only the first time you use them. Once they are covered with a tile, the double or triple no longer applies(say, if on a later move, you add an S to the word)
Yes I did think of that and even looked at the official Scrabble web site. The question came up in a discussion with a co-worker. I’ve always played as to allow the example n the OP, but he read the instructions differently. He knew that things like adding “emon” below were allowed:
p
lemon
a
y
But in his mind the specific example in the OP was two separate plays: one to pluralize the existing word and another to build a new word at right angles.
BTW, there is nothing in the OP about scoring.
No, it’s one play that forms two separate (for scoring purposes) words. Anything that involves laying down exactly one continuous line - horizontal or vertical - of letters is one play. The number of new words formed, be it one, two, three, or more, does not define the number of plays.
Yes, Scrabble would be pretty dull (not to say hard, at later stages of the game) if you could only make one new word per play. Often the highest scoring move involves a fairly mundane word but with lots and lots of crossing letters. With suitable words you can make 5 or 6 new words in one play.
e.g. you have this board:
M
O
TORQUE
R
NIP
STAR
with just four letters you can make 5 new words:
M
O
TORQUE
**WEI**R**D**
NIP
STAR
Not meant to be snarky, but. . .
Does your game have a boxtop with the rules on it? They give examples at the bottom, and I’m pretty sure that one of the examples shown is like your example.