We present results of Monte Carlo study of the monomer-monomer correlation functions, static structure factor, and asphericity characteristics of a single homopolymer in the coil and globular states for three distinct architectures of the chain: ring, open, and star. To rationalize the results we introduce the dimensionless correlation functions rescaled via the corresponding mean-squared distances between monomers. For flexible chains with some architectures these functions exhibit a large degree of universality by falling onto a single or several distinct master curves. In the repulsive regime, where a stretched exponential times a power law form (de Cloizeaux scaling) can be applied, the corresponding exponents delta and theta have been obtained. The exponent delta=1/nu is found to be universal for flexible strongly repulsive coils and in agreement with the theoretical prediction from improved higher-order Borel-resummed renormalization group calculations. The short-distance exponents theta(v) of an open flexible chain are in a good agreement with the theoretical predictions in the strongly repulsive regime also. However, increasing the Kuhn length in relation to the monomer size leads to their fast crossover towards the Gaussian behavior. Likewise, a strong sensitivity of various exponents theta(ij) on the stiffness of the chain, or on the number of arms in star polymers, is observed. The correlation functions in the globular state are found to have a more complicated oscillating behavior and their degree of universality has been reviewed. Average shapes of the polymers in terms of the asphericity characteristics, as well as the universal behavior in the static structure factors, have been also investigated.