Recently, a new way to amplify DNA, called solid phase amplification (SPA), has been introduced. SPA differs from the traditional polymerase chain reaction (PCR) in the use of surface-bound instead of freely-diffusing primers to amplify DNA. This limits the amplification to two-dimensional surfaces and therefore allows the easy parallelization of DNA amplification in a single system. Furthermore, SPA could provide an alternate route to DNA target implantation on DNA chips for genomic studies. Standard PCR processes are usually characterized (at least initially) by an exponential growth and a broad population distribution, and they are well described by the theory of branching processes, wherein a generating function can be used to obtain the probability distribution function for the population of offspring. This theoretical approach is not appropriate for SPA because it cannot properly take into account the many-body (steric) and geometric effects in a quenched two-dimensional environment. In this article, we propose a simple Lattice Monte Carlo technique to model SPA. We study the growth, stability, and morphology of isolated DNA colonies under various conditions. Our results indicate that, in most cases, SPA is characterized by a geometric growth and a rather sharp size distribution. Various non-ideal effects are studied, and we demonstrate that such effects do not generally change the nature of the process, except in extreme cases.