In this paper, we consider the problem of designing Differentially Private (DP) algorithms for Stochastic Convex Optimization (SCO) on heavy-tailed data. The irregularity of such data violates some key assumptions used in almost all existing DP-SCO and DP-ERM methods, resulting in failure to provide the DP guarantees. To better understand this type of challenges, we provide in this paper a comprehensive study of DP-SCO under various settings. First, we consider the case where the loss function is strongly convex and smooth. For this case, we propose a method based on the sample-and-aggregate framework, which has an excess population risk of Õ(d^3/nϵ^4) (after omitting other factors), where n is the sample size and d is the dimensionality of the data. Then, we show that with some additional assumptions on the loss functions, it is possible to reduce the expected excess population risk to Õ( d^2/ nϵ^2 ). To lift these additional conditions, we also provide a gradient smoothing and trimming based scheme to achieve excess population risks of Õ( d^2/nϵ^2) and Õ(d^2/3/(nϵ^2)^1/3) for strongly convex and general convex loss functions, respectively, with high probability. Experiments on both synthetic and real-world datasets suggest that our algorithms can effectively deal with the challenges caused by data irregularity.