At least 30% of main sequence stars host planets with sizes between 1 and 4 Earth radii and orbital periods of less than 100 days. We use N-body simulations including a model for gas-assisted pebble accretion and disk-planet tidal interaction to study the formation of super-Earth systems. We show that the integrated pebble mass reservoir creates a bifurcation between hot super-Earths or hot-Neptunes ($\lesssim15M_{\oplus}$) and super-massive planetary cores potentially able to become gas giant planets ($\gtrsim15M_{\oplus}$). Simulations with moderate pebble fluxes grow multiple super-Earth-mass planets that migrate inwards and pile up at the disk's inner edge forming long resonant chains. We follow the long-term dynamical evolution of these systems and use the period ratio distribution of observed planet-pairs to constrain our model. Up to $\sim$95% of resonant chains become dynamically unstable after the gas disk dispersal, leading to a phase of late collisions that breaks the resonant configuration. Our simulations match observations if we combine a dominant fraction ($\gtrsim95\%$) of unstable systems with a sprinkling ($\lesssim5\%$) of stable resonant chains (the Trappist-1 system represents one such example). Our results demonstrate that super-Earth systems are inherently multiple (${\rm N\geq2}$) and that the observed excess of single-planet transits is a consequence of the mutual inclinations excited by the planet-planet instability. In simulations in which planetary seeds are initially distributed in the inner and outer disk, close-in super-Earths are systematically ice-rich. This contrasts with the interpretation that most super-Earths are rocky based on bulk density measurements of super-Earths and photo-evaporation modeling of their bimodal radius distribution. We investigate the conditions needed to form rocky super-Earths. The formation of rocky super-Earths (abridged)Read less <