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I have some tables of numbers, most of which are small (0, 1, -1), but occasionally I have fractions. If I just has 0, 1, -1's I'd right align so the $-$ signs will line up, but the fractions throw things off. Center align doesn't look great either, nor does left align. Here is a small example of a right and center align:

begintabularrr
$frac 124$ & $-frac 124$ \
$1$ & $-1$ \
$-1$ & $0$ \
endtabular
quad
begintabularcc
$frac 124$ & $-frac 124$ \
$1$ & $-1$ \
$-1$ & $0$ \
endtabular

Can I center align the numbers without the signs, so the top two rows will look like they do on the right in the example above, and the bottom two rows will look like the do on the left in the above example?

With phantom you can insert an invisible -:

documentclassarticle

begindocument
begintabularcc
$phantom-frac 124$ & $-frac 124$ \
$phantom-1$ & $-1$ \
$-1$ & $phantom-0$ \
endtabular
enddocument

If your table contains only (mostly) math, you might want to use array instead of tabular.

documentclassarticle

usepackagemathtools

begindocument

begintabularcc
$frac 124$ & $mathllap-frac 124$ \
$1$ & $mathllap-1$ \
$mathllap-1$ & $0$ \
endtabular

enddocument

Here I make - active (inside a limited group) and define it as a mathllap. I use a TABstack with enough column separation to allow for the -.

documentclassarticle
usepackagetabstackengine,mathtools
TABstackMath
begindocument
bgroup
setstackgapL16pt
setstacktabbedgap13pt
letsvm-
catcode`-=active
def-mathllapsvm
Matrixstackfrac 124 & -frac 124 \ 1 & -1 \ -1 & 0
quadMatrixstackfrac 124 & -frac 124 \ 1 & -1 \ -1 & 0 
egroup
enddocument