I have deposited the thin film of rare earth transition metal oxide on LAO and STO (100) substrate and taken the magnetization of the films (M vs T @100 Oe from 10K to 300K and M vs H @10 K). Now I want to analysis the data.
If the substrate is paramagnetic within the temperature range then its paramagnetic contribution may be removed using analytical form of Curie law M(T) ~ 1/T (variable temperature measurements). Correction for measurements in changing magnetic field may be corrected as M'(H) = M(H) - a*H, with appropriate value of parameter a (a > 0) - at least in not too high fields. At lower temperatures the term a*H should be rather replaced with Langevin function L(H,T). Using analytical models leaves experimental uncertainties on their original level, while the method described by E. Kulakov generally increases the noises.
In M(H) measurement things are quite easy: you have to subtract a straight line with negative slope from original data (fixed and negative susceptibility). But when superconductivity might be at play then the procedure complicates a bit and depends whether we are playing with type-I or type-II superconductor. Hopefully it is not your case, as the lowest temperature (10K) seems too high. I'm not sure what about M(T) measurements with diamagnetic part.
I never did it myself (subtraction of paramagnetic part from M(T) curve. But look at Curie law: magnetic susceptibility is proportional to 1/(T-TC) for T>TC, where TC is Curie temperature of a given paramagnet. Exactly the same behavior is thus expected for its M(T) curve. This approach is an idealization and might appear tricky in practical situation, even if your paramagnetic background is generated by exactly one chemical compound, with known Curie temperature. The plot of M(T) vs. 1/T may be helpful - purely paramagnetic part should appear as a straight line. But certainly the approach already recommended by E. Kulakov should be preferred.