We consider a two-class clustering model: Xi = liμ+Zi for 1 ≤ i ≤ n, where li ∈{−1, 1} are the unknown class label, μ ∈R is a sparse vector and Zi ∼N(0,Ip). In this model, we study three interconnected problems: (1) global testing, where H0 : Xi = Zi, 1 ≤ i ≤ p versus H1 being the two-class model above; (2) clustering (estimating the label vector l); (3) feature selection (recovering the support of μ). Under a Rare/Weak signal model, we show fundamental statistical limits for the three desired goals and deliver sharp phase diagrams. We propose a method, Important Features (IF)-PCA, for high-dimensional clustering. Classical PCA is a standard method for clustering, but it faces challenges when p is much larger than n. IF-PCA has two major innovations: (1) It uses chi-square screening to remove useless features in PCA; (2) It uses an adaption of the recently developed Higher Criticism Thresholding (HCT) for deciding the threshold in chi-square screening. The method is fast in computation and is tuning-free. We investigate the Hamming clustering error of IF-PCA under Rare/Weak signal model, where more subtle phase transition phenomenon is revealed in a delicate range of signal strength. We apply IF-PCA to 10 gene microarray data sets. In several of these data sets, the method yields much lower error rate than other popular clustering methods including classical PCA, k-means algorithm and hierarchical clustering.