Using maximum likelihood, any of these can be compared with AIC, and any that are nested can be compared with the likelihood ratio test. Using REML, the fixed effects must be the same; you have to use ML to compare models with different fixed effects. However, interpretation is usually difficult when both fixed effects and random effects are changing, so in practice, most recommend changing only one or the other at a time.

Pinhiero/Bates is the classic reference; it describe the nlme package, but the theory is the same. Well, mostly the same; Doug Bates has changed his recommendations on inference since writing that book and the new recommendations are reflected in the lme4 package. But that's more than I want to get into here. A more readable reference is Weiss (2005), Modeling Longitudinal Data.

Using maximum likelihood, any of these can be compared with AIC; if the fixed effects are the same (m1 to m4), using either REML or ML is fine, with REML usually preferred, but if they are different, only ML can be used. However, interpretation is usually difficult when both fixed effects and random effects are changing, so in practice, most recommend changing only one or the other at a time.

Using the likelihood ratio test is possible but messy because the usual chi-squared approximation doesn't hold when testing if a variance component is zero. See Aniko's answer for details. (Kudos to Aniko for both reading the question more carefully than I did and reading my original answer carefully enough to notice that it missed this point. Thanks!)

Pinhiero/Bates is the classic reference; it describes the nlme package, but the theory is the same. Well, mostly the same; Doug Bates has changed his recommendations on inference since writing that book and the new recommendations are reflected in the lme4 package. But that's more than I want to get into here. A more readable reference is Weiss (2005), Modeling Longitudinal Data.

Using maximum likelihood, any of these can be compared with either the likelihood ratio test or AIC. Using REML, the fixed effects must be the same; so m5 is not comparable to the others. However, interpretation is usually difficult when both fixed effects and random effects are changing, so in practice, most recommend changing only one or the other at a time.

Pinhiero/Bates is the classic reference; it describe the nlme package, but the theory is the same. Well, mostly the same; Doug Bates has changed his recommendations on inference since writing that book and the new recommendations are reflected in the lme4 package. But that's more than I want to get into here. A more readable reference is Weiss (2005), Modeling Longitudinal Data.

Using maximum likelihood, any of these can be compared with AIC, and any that are nested can be compared with the likelihood ratio test. Using REML, the fixed effects must be the same; you have to use ML to compare models with different fixed effects. However, interpretation is usually difficult when both fixed effects and random effects are changing, so in practice, most recommend changing only one or the other at a time.

Pinhiero/Bates is the classic reference; it describe the nlme package, but the theory is the same. Well, mostly the same; Doug Bates has changed his recommendations on inference since writing that book and the new recommendations are reflected in the lme4 package. But that's more than I want to get into here. A more readable reference is Weiss (2005), Modeling Longitudinal Data.