Since your t has 5 items and each column in your s has 5 items I am guessing you want to plot each column of s using the values of t on your horizontal axis
Mathematica usually deals with rows so I am going to use Transpose to exchange your rows and columns of s.
ListPlot uses 1,2,3,4,5 for the horizontal axis unless you have pairs of numbers like {{-1.,1},{-0.5,4},{0.,7},{0.5,10},{1.,13}}
So I am going to write a little function that will pair up items from t along with individual items from columns in s. That function will again use Transpose to pair up {-1.,-0.5,0.,0.5,1.} and {1,4,7,10,13} and turn that into {{-1.,1},{-0.5,4},{0.,7},{0.5,10},{1.,13}} for the first column and do the same for the rest of your columns. And, by making all these changes to the data, the labels on the x axis of your plot match the values in your list t.
Try
s={{1,2,3},{4,5,6},{7,8,9},{10,11,12},{13,14,15}};
t=Range[-1,1,0.5];
f[v_]:=Transpose[{t,v}];
ss=Map[f,Transpose[s]];
ListPlot[ss]
and see if my guesses are correct. I realize that may seem to be complicated just to get the labels on the x axis.
