In this paper, we are aiming to prove several regularity results for the following stochastic fractional heat equations with additive noisesdut(x)=Δα2ut(x)dt+g(t,x)dηt,u0=0,t∈(0,T],x∈G, for a random field u:(t,x)∈[0,T]×G↦u(t,x)=:ut(x)∈R, where Δα2:=−(−Δ)α2,α∈(0,2], is the fractional Laplacian, T∈(0,∞) is arbitrarily fixed, G⊂Rd is a bounded domain, g:[0,T]×G×Ω→R is a joint measurable coefficient, and ηt,t∈[0,∞), is either a Brownian motion or a Lévy process on a given filtered probability space (Ω,F,P;{Ft}t∈[0,T]). To this end, we derive the BMO estimates and Morrey–Campanato estimates, respectively, for stochastic singular integral operators arising from the equations concerned. Then, by utilizing the embedding theory between the Campanato space and the Hölder space, we establish the controllability of the norm of the space Cθ,θ/2(D¯), where θ≥0,D¯=[0,T]×G¯. With all these in hand, we are able to show that the q-th order BMO quasi-norm of the αq0-order derivative of the solution u is controlled by the norm of g under the condition that ηt is a Lévy process. Finally, we derive the Schauder estimate for the p-moments of the solution of the above stochastic fractional heat equations driven by Lévy noise.