Trick for writing reg exp :-
The Regular expression for the set of all binary strings with an even number of $1$′$s$:-
$(0^*10^*10^*)^*+0^*$
$\ $
The Regular expressions for the set of all binary strings with an odd number of $1$′$s$:-
1. Remove last $1$, whatever remains is even no. of $1$’$s$, then place $1$ at last
$((0^*10^*10^*)^*+0^*)10^*$
OR
2. Take first $1$ out, whatever remains is even no. of $1$’$s$
$0^*1((0^*10^*10^*)^*+0^*)$