When should I use non-breaking space?

To quote Knuth, ties should appear:

• In references to named parts of a document:
  Chapter~12   Theorem~1.2
Appendix~A   Table~\hbox{B-8}
Figure~3   Lemmas 5 and~6.
• Between a person's forenames and between multiple surnames:
  Donald~E. Knuth   Luis~I. Trabb~Pardo
Bartel~Leendert van~der~Waerden   Charles~XII
  but be careful of names like Charles Louis Xavier~Joseph de~la Vall\'ee~Poussin.
• Between math symbols in apposition with nouns:
  dimension~$d$ width~$w$ function~$f(x)$
string~$s$ of length~$l$~
  but compare with
  string~$s$ of length $l$~or more.
• Between symbols in series:
  1,~2, or~3
$a$,~$b$, and~$c$
1,~2, \dots,~$n$.
• When a symbol is a tightly bound object of a preposition:
  of~$x$
from 0 to~1
increase $z$ by~1
in common with~$m$. but compare
  of $u$~and~$v$.
• When mathematical phrases are rendered in words:
  equals~$n$ less than~$\epsilon$ (given~$X$)
mod~2 modulo~$p^e$ for all large~$n$
  Compare is~15 with is 15~times the height.
• When cases are being enumerated within a paragraph:
  (b)~Show that $f(x)$ is (1)~continuous; (2)~bounded..

In general where the break will create orphans that would distract the reader.

Some less obvious examples:

I~am
I~definitely
mod~1


The matching $(AW,BX,CY,DZ)$ is unstable, because for example
$A$ prefers~$Z$ to~$W$ and at the same time $Z$ prefers~$A$ to~$D$.
But the matching $(AZ,BW,CX,DY)$ is stable;

(we say that girl~$h$ rejects the proposal)

step~A2 stops when $P$ has nobody left to propose~to,
but step~B2 keeps making redundant proposals ad~infinitum when

The details of Algorithm~B

has local probability~${1\over n}$,

The "I am", "I definitely" etc., is a bit controversial, but personally like a lot of other people don't like "I" at the end of a line break.

In languages like German where spaced en dashes are used for sentence insertions, it is often frowned upon to place the en dash at the start of a new line. So, a non-breaking space should be placed before the dash.

Dieses Mal~-- anders als vorher~-- wurde er überrascht.

This time---unlike before---he was caught by surprise.