We present high-resolution measurements of the coefficient of thermal expansion ␣(T)ϭץ ln l(T)/ץT of the quasi-two-dimensional ͑quasi-2D͒ salts -(BEDT-TTF) 2 X with XϭCu(NCS) 2 , Cu͓N(CN) 2 ͔Br, and Cu͓N(CN) 2 ͔Cl in the temperature range Tр150 K. Three distinct kinds of anomalies corresponding to different temperature ranges have been identified. These are ͑A͒ phase-transition anomalies into the superconducting "XϭCu(NCS) 2 , Cu͓N(CN) 2 ͔Br… and antiferromagnetic "XϭCu͓N(CN) 2 ͔Cl… ground state, ͑B͒ phasetransition-like anomalies at intermediate temperatures ͑30-50͒ K for the superconducting salts, and ͑C͒ kinetic, glasslike transitions at higher temperatures, i.e., ͑70-80͒ K for all compounds. By a thermodynamic analysis of the discontinuities at the second-order phase transitions that characterize the ground state of system ͑A͒, the uniaxial-pressure coefficients of the respective transition temperatures could be determined. We find that in contrast to what has been frequently assumed, the intraplane-pressure coefficients of T c for this family of quasi-2D superconductors do not reveal a simple form of systematics. This demonstrates that attempts to model these systems by solely considering in-plane electronic parameters are not appropriate. At intermediate temperatures ͑B͒, distinct anomalies reminiscent of second-order phase transitions have been found at T* ϭ38 K and 45 K for the superconducting XϭCu(NCS) 2 and Cu͓N(CN) 2 ͔Br salts, respectively. Most interestingly, we find that the signs of the uniaxial pressure coefficients of T*, ץT*/ץp i (iϭa,b,c), are strictly anticorrelated with those of T c . Based on comparative studies including the nonsuperconducting X ϭCu͓N(CN) 2 ͔Cl salt as well as isotopically labeled compounds, we propose that T* marks the transition to a density-wave state forming on minor, quasi-1D parts of the Fermi surface. Our results are compatible with two competing order parameters that form on disjunct portions of the Fermi surface. At elevated temperatures ͑C͒, all compounds show ␣(T) anomalies that can be identified with a kinetic, glasslike transition where, below a characteristic temperature T g , disorder in the orientational degrees of freedom of the terminal ethylene groups becomes frozen in. Our results provide a natural explanation for the unusual time-and cooling-rate dependences of the ground-state properties in the hydrogenated and deuterated Cu͓N(CN) 2 ͔Br salts reported in the literature.