Fitting a smooth curve to 2D, a surface to 3D, and a manifold to 4D irregular point cloud data is becoming a common practice in many engineering and science applications. Piecewise-polynomial spline functions provide a powerful tool applicable to interpolation and approximation problems. This study presents the least squares B-spline approximation (LSBSA) theory, which is a generalized version of the spline interpolation and can be applied to any irregularly scattered point cloud data at knots specified by the user. The formulation allows to apply the well-established body of knowledge of least squares theory to the B-spline approximation. This for example has the benefit of embedding quality control measures such as hypothesis testing and proper error propagation to assess the quality of the approximation problem. The method is applicable to many 1D curve, 2D surface and 3D manifold fitting problems of which both simulated and real data are used to illustrate the efficacy of the proposed theory. In particular, its real-world applications to multi-beam echo-sounder bathymetric data, digital terrain modeling and Greenland ice sheet deformation monitoring will be highlighted. The performance of the method for linear, quadratic, cubic and quartic spline functions will be investigated. The primary application of LSBSA lies in its ability to perform 3D manifold fitting for deformation monitoring. This capability provides the possibility of monitoring changes in continuous spatial and temporal domains. The Python and Matlab source codes of LSBSA are freely accessible at https://github.com/tudelft4d/lsbsa.