If you live near a high-voltage power line, that may well be the dominant source of magnetic field in your home.  See various graphs that you can use to see how near you have to be for this to be the case, but typically, we are talking a hundred metres or less rather than hundreds of metres.

Measures that only the power company could take

Changing the design of the line

The field produced by a power line depends on many factors, but it is unlikely that the electricity companies would change any of these for an existing line in order to reduce the fields.  However the UK does have a policy of optimal phasing - this is a measure that can be applied to some power lines that reduces the field.  Power companies in the UK normally apply this to new lines and have agreed a programme of retrofitting in some circumstances to existing lines that do not already have it.

Undergrounding

When a high-voltage power line is buried, the magnetic field changes.  It can actually be higher directly above the underground cables.  But the range of the field reduces, so in practice, it would usually reduce the exposures in nearby homes.  However, power companies in the UK do not bury high-voltage power lines just for EMF reasons.  See more on undergrounding on National Grid's website.

Measures that an individual could take

Screening the field

It is possible to screen magnetic fields but this involves large thicknesses (typically a centimetre or more) of metal all round the area being screened, is expensive, and is rarely applied in domestic settings.  Normal house building materials already provide a large degree of screening against electric fields.  In principle, it is also possible to build an active cancellation system, but this is tricky, and again, to our knowledge, has almost never been done in domestic settings in the UK.

Distance

Other than these, the only real way of reducing exposures from a high-voltage power line is to increase the distance from it.

The commonest source of magnetic field inside a home is the low-voltage distribution wiring outside the home supplying power to homes along a street.  These circuits produce magnetic fields mainly because of net currents. 

Measures that only the power company could take

SAGE considered net currents and how they arise, and concluded that it is practically impossible to eliminate them without compromising safety.  But there are some "best practice" measures, most of which power companies already take, that can reduce the chances of large net currents.  Sometimes (but by no means every time), high magnetic fields in a home can be a sign of a fault on the distribution network.  The electricity company would like to know about this, so if you suspect this is the case, perhaps because you have flickering lights as well, you can draw it to their attention.

Measures that an individual could take

In most cases, net currents are produced by features of the distribution circuits, and there is little a householder can do to reduce them.  But sometimes, a house may have a higher-than-usual magnetic field because a net current enters the house through a conducting water or gas pipe.  SAGE discusses options of fitting a plastic section to water pipes entering the home or of fitting inductors to the electricity intake.  Note that both of these have other safety implications and must be performed by qualified professionals only.

Normal wiring

Most normal house wiring is not a major source of magnetic field.  It often is a source of electric fields.  If it is desired to reduce electric fields, options are using screened wiring (a fairly radical departure from normal practice) or even disconnecting the voltage from circuits when not in use.  See  manufacturers of such equipment. 

Faulty or incorrect wiring

Sometimes, a particular feature of the house wiring becomes a source of magnetic field, e.g. two-way switching of lights, or a faulty ring-main circuit.  SAGE produced a good treatment of these features and how to identify and eliminate them.  This includes options of fitting residual current breakers and using radial circuits instead of ring mains.  Once again, these options must not be allowed to jeopardise safety or contravene wiring regulations, and must be performed only by qualified professionals.

Design of appliance

With some types of appliances, new designs produce lower fields than old designs.  For example, modern switched-mode power supplies produce lower fields than older transformer-based power supplies for chargers, adaptors, etc.

Distance from the appliance

For virtually all appliances, the field falls quite rapidly with distance, usually dropping off over the first metre or less.  So the easy way to avoid exposure from appliances is not to get too close to them.  This applies particularly to appliances on e.g. the bedside table, where the exposure could be for an extended period of time.

Switching off the appliance

To reduce, specifically, electric fields, turn the appliance off at the mains socket rather than just at the appliance's own switch.  This stops the wire connecting the appliance from producing electric fields.  This makes no difference to magnetic fields, which are produced only when the appliance is operating.