  We use the new Modular Open Source Fitter for Transients (MOSFiT) to model 38
hydrogen-poor superluminous supernovae (SLSNe). We fit their multicolour light
curves with a magnetar spin-down model and present the posterior distributions
of magnetar and ejecta parameters. The colour evolution can be well matched
with a simple absorbed blackbody. We find the following medians (1$\sigma$
ranges): spin period 2.4 ms (1.2-4 ms); magnetic field $0.8\times 10^{14}$ G
(0.2-1.8 $\times 10^{14}$ G); ejecta mass 4.8 Msun (2.2-12.9 Msun); kinetic
energy $3.9\times 10^{51}$ erg (1.9-9.8 $\times 10^{51}$ erg). This
significantly narrows the parameter space compared to our priors, showing that
although the model is flexible, the parameter space relevant to SLSNe is well
constrained by existing data. The requirement that the instantaneous engine
power is $\sim 10^{44}$ erg at the light curve peak necessitates either a large
rotational energy (P<2 ms), or more commonly that the spin-down and diffusion
timescales be well-matched. We find no evidence for separate populations of
fast- and slow-declining SLSNe, which instead form a continuum both in light
curve widths and inferred parameters. Variations in the spectra are well
explained through differences in spin-down power and photospheric radii at
maximum-light. We find no correlations between any model parameters and the
properties of SLSN host galaxies. Comparing our posteriors to stellar evolution
models, we show that SLSNe require rapidly rotating (fastest 10%) massive stars
(> 20 Msun), and that this is consistent with the observed SLSN rate. High
mass, low metallicity, and likely binary interaction all serve to maintain
rapid rotation essential for magnetar formation. By reproducing the full set of
SLSN light curves, our posteriors can be used to inform photometric searches
for SLSNe in future survey data.
