Strong methylating agents generally tend to be carcinogenic. Substitutes are less carcinogenic if they are less powerful methylating agents. Often powerful methylating agents are needed. So it will not work to perform every methylation with dimethyl carbonate.
Therefore it is a matter to balance the concrete synthetic problem. In this respect intention should also paid to the question if it is meaningful to exchange "sure"-carcinigenic dimethyl sulfate by iodomethane which is only suspected to be carcinogenic but highly volatile (bp. = 42 °C).
Alkylsulfonates are well known alkylating agents. Most known are tosylates (4-toluenesulfonates). Furthermore there are also brosylates (4-bromobenzenesulfonates), nosylates (4-nitrobenzenenesulfonates) and mesylates (methanesulfonates). However these compounds have no 'harmonised classification'. Therefore the classifications of the hazardous potential may vary from manufacturer to manufacturer.
Methyltosylate is easily obtained. The price is a little bit lower than of iodo methane if molar amounts are compared. Above all it has a much higher boiling point (292 °C) and only moderate toxic potential. From a security standpoint it is therefore far superior to iodomethane.
Methylmethansulfonate is also carcinogenic. From a security standpoint it is therefore no suitable substitute. Other methylsulfonates are difficult to get, have a high price or are commercially unavailable.
Some years ago dimethylcarbonate has been advertised as a suitable substitute. Dimethylcarbonate is even a little bit cheaper than dimethylsulfate. But the alkylating power is only poor. Therefore rigorous reaction conditions are needed and there are substrates which will not react at all. You have to search the literature to find out what will work. If you cannot get any information, you can try it out. Since the job is to perform the reaction with less hazardous compounds this is a good task for students in the lab courses.