Cheat Code: spherical Bessel roots are their respective half-integer Bessel roots
According to Abramowitz, 1964, Ch9, pp 440,"Zeros and Their Asymptotic Expansions"
The zeros of $j_{n}(x)$ and $y_{n}(x)$ [i.e.: Spherical Bessel] are the same as $J_{n+1/2}(x)$ and $N_{n+1/2}(x) $ [i.e.: 'Normal' Bessel]
Mathematica Implementation
I used, for the spherical Bessel of the first kind:
nRoots = 4; nBessel = 3; SphBesselRoot[l_, k_] := N[BesselJZero[l + 1/2, k]]; Grid[Table[SphBesselRoot[l, i], {l, 0, nBessel}, {i, 1, nRoots}]]
This prints a table of the first four roots (columns) for the four spherical Bessel functions (rows) :
3.14159 6.28319 9.42478 12.5664 4.49341 7.72525 10.9041 14.0662 5.76346 9.09501 12.3229 15.5146 6.98793 10.4171 13.698 16.9236
Comparing to tables of the roots online, this table is in good agreement.
I found this method easier than root finding or asymptotic expressions.