A tankless water heater is much more efficient than a traditional water heater because it only heats the water that is being used instead of heating and reheating water stored in a tank. Rinnai, Rheem, and Bosch all make versions of tankless water heaters that are well-reviewed.
One of the most energy efficient water heating methods is the heat pump water heater. This method is proven to be two times more efficient over the standard electric water heater.
Any heater that has the Energy Star label will be very energy efficient. The Honeywell eco line, for instance, are very energy efficient.
The most energy efficient radiator heater is the oil filled one. A better option would be an electrical one because those are almost 100% energy efficient.
The question is better defined as "What the most energy efficient portable heater?" Because one would assume that you will want to be able to take the heater with you when or if you move.
A large energy saving one. It will keep lots of hot water and will use less energy when not in use.
instant gas
There is no single answer to this question. Water heaters with Energy Star certification are your best bet. However, the important considerations are what you need. If you get the right size, set the right temperature, and insulate your water heater, that will make more difference than the specific brand. The most energy efficient hot water heaters are tankless. Check your local Home Depot or Lowes for pricing and installation information.
Gas-fired, storage-based water pumps are the most energy efficient type.
Vornado is the brand of heater that is the most energy efficient. If used properly, Vornado heaters can heat just as well as a larger heater, for a much lower cost. They are small, portable, and can heat large rooms. Some of them even have thermostats or timers to increas efficiency.
I have no experience in this, however A. O Smith claims that there brand of gas water heaters are the most efficient. They also offer a hybrid water heater.
The Fahrenheat Electric 5120 BTU Baseboard Heater is very energy efficient. Because of its "baseboard" design, it is not necessary to use a powerful to disperse the heat. It uses natural radiance, which saves on electricity costs.
Indirect water heaters are one of the most energy efficient, low cost water heaters on the market today. It stores 40 gallons of water for the cost of $0.05 per day.