The book TeX By Topic by Victor Eijkhout describes some \show... macros.
In the following overview X means some integer value and Y and 8-bit number, \Z is short for some 'internal quantity' such as a count, dimen, \skip or token register, W is a word or space separated list of words.
\show is primitive command and shows the meaning of a token, i.e. what \meaning\foo does within the document.
\showboxdepth=X has the value of the deepest level of box nesting, defaulting to X=3 in plain TeX
\showboxbreadth=X has the maximum number of items shown per level
This has a plain TeX default of X=5
\showlists is a primitive and writes the content of partial box lists in all of the 4 non-math TeX modes (i.e. horizontal/vertical and the restricted versions)
\showthe\Y prints the content of a \Y quantity as in the document with \the, \the\footoks
\showhyphens{W} displays the hyphenation of W on the terminal/log according to the hyphenation rules.
Now, the eTeX standard (which is a standard by now as of writing this in April 2017) adds some more \show... tools
\showifs (activated with \tracingifs=1)
\showgroups (\tracinggroups=1)
\showtokens for token lists and content, that is not possible to show with \show or \showthe.
These three eTeX additions are mode-independent.
LaTeX adds some wrapper macros, such as
\showoutput
\showoverfull
\showhyphens (being redefined)
and sets initially \showboxbreadth=-1 and \showboxdepth=-1 as well.
Here's a sample LaTeX document for the plain TeX \show... entities.
\documentclass{article}
\newbox\foobox
\setbox\foobox=\hbox{A nice text}
\newcount\foocnt
\foocnt=17
\newtoks\footok
\footok={Gandalf The Grey}
\tracingoutput=1
\tracingonline=1
\begin{document}
\showboxbreadth=10
\showboxdepth=2
\scrollmode
\showbox\foobox
\showthe\footok
\showthe\foocnt
% Show lists
a\hfil\break b\par
c\hfill\break d
\hbox{e\vbox{f\showlists}}
\end{document}
The output is both on console as in the .log file (here I cut the header and the bottom statistics to provide the \show... related output only):
> \box26=
\hbox(6.83331+0.0)x48.88896
.\OT1/cmr/m/n/10 A
.\glue 3.33333 plus 1.66498 minus 1.11221
.\OT1/cmr/m/n/10 n
.\OT1/cmr/m/n/10 i
.\OT1/cmr/m/n/10 c
.\OT1/cmr/m/n/10 e
.\glue 3.33333 plus 1.66666 minus 1.11111
.\OT1/cmr/m/n/10 t
.\OT1/cmr/m/n/10 e
.\OT1/cmr/m/n/10 x
.etc.
! OK.
l.23 \showbox\foobox
> Gandalf The Grey.
l.25 \showthe\footok
> 17.
l.27 \showthe\foocnt
### horizontal mode entered at line 32
\hbox(0.0+0.0)x15.0
\OT1/cmr/m/n/10 f
spacefactor 1000
### internal vertical mode entered at line 32
prevdepth ignored
### restricted horizontal mode entered at line 32
\OT1/cmr/m/n/10 e
spacefactor 1000
### horizontal mode entered at line 31
\hbox(0.0+0.0)x15.0
\OT1/cmr/m/n/10 c
\glue 0.0 plus 1.0fill
\penalty -10000
\OT1/cmr/m/n/10 d
\glue 3.33333 plus 1.66666 minus 1.11111
spacefactor 1000
### vertical mode entered at line 0
### current page:
\write-{}
\glue(\topskip) 5.69446
\hbox(4.30554+0.0)x345.0, glue set 324.99998fil
.\hbox(0.0+0.0)x15.0
.\OT1/cmr/m/n/10 a
.\glue 0.0 plus 1.0fil
.\penalty -10000
.\glue(\rightskip) 0.0
\penalty 300
\glue(\baselineskip) 5.05556
\hbox(6.94444+0.0)x345.0, glue set 339.44443fil
.\OT1/cmr/m/n/10 b
.\penalty 10000
.\glue(\parfillskip) 0.0 plus 1.0fil
.\glue(\rightskip) 0.0
\glue(\parskip) 0.0 plus 1.0
total height 22.0 plus 1.0
goal height 550.0
prevdepth 0.0
! OK.
l.32 \hbox{e\vbox{f\showlists
}}
Overfull \hbox (12.22223pt too wide) in paragraph at lines 31--33
\OT1/cmr/m/n/10 d []
\hbox(6.94444+0.0)x345.0, glue set - 1.0
.\OT1/cmr/m/n/10 d
.\glue 3.33333 plus 1.66666 minus 1.11111
.\hbox(6.94444+0.0)x349.44444
..\OT1/cmr/m/n/10 e
..\vbox(6.94444+0.0)x345.0 []
.\penalty 10000
.\glue(\parfillskip) 0.0 plus 1.0fil
.\glue(\rightskip) 0.0
Completed box being shipped out [1]
\vbox(633.0+0.0)x407.0
.\glue 16.0
.\vbox(617.0+0.0)x345.0, shifted 62.0
..\vbox(12.0+0.0)x345.0, glue set 12.0fil []
..\glue 25.0
..\glue(\lineskip) 0.0
..\vbox(550.0+0.0)x345.0, glue set 503.94617fil []
..\glue(\baselineskip) 23.55556
..\hbox(6.44444+0.0)x345.0, glue set 170.0fil []
texdoc etex) adds a few more. there are also some convenience latex macros starting with show such as\showoutput(or rather etex adds more\tracing...commands) – David Carlisle Apr 16 '17 at 12:29