Be careful with wood fireplaces — most emit A LOT of very bad particulate matter!
Burning wood in a common type of stove may result in more particulate matter than a diesel truck and may carry more carcinogens than a diesel or petrol exhaust. In the United Kingdom, where many people have such stoves or even open hearths, wood burning generates twice as much PM2.5 as all traffic combined. In many urban areas, burning wood is prohibited (I don't know if those bylaws make exceptions for modern, cleaner stoves). Globally, biomass is estimated to be responsible for 250,000 casualties per year, as the overwhelming majority of biomass burning takes place with open fire or primitive stoves.
From an environmental point of view, burning solid fuel at home may be much worse than you think, depending on the type of fireplace you have. Wood may be (at least potentially) be carbon neutral, that does not mean the exhaust is anywhere near healthy to breathe. Check the characteristics of your fireplace before you assume that wood is carbon neutral and therefore environmentally friendly.
See this article in The Guardian:
Wood smoke is thick with the tiny particulates, known as PM2.5, that are linked to heart attacks, strokes, cancer, dementia and various other ailments. What’s more, the claims about a climate benefit from wood use are questionable.
Cars and trucks get more attention but nationally, domestic wood burning is the largest single source of PM2.5. According to one analysis of government data, it produces more than twice as much as all road traffic. While concerns about diesel vehicles focus largely on the nitrogen dioxide they produce, the evidence tying particulates to death and disease is even more powerful.
According to Leigh Crilley, an atmospheric chemist at the University of Birmingham, wood smoke also carries more carcinogens than diesel or petrol exhaust.
The increasing popularity of wood fires, scientists warn, threatens to erase any progress big cities might achieve in reducing pollution from traffic. “It’s overtaking the gains we’re making,” Crilley said.
One study from 2014 found that wood smoke was adding more particle pollution to London’s air than the first two phases of the city’s low-emission zone were expected to remove. In London and Birmingham, King’s College researchers reported wood accounted for up to 31% of locally produced particulates. And across Europe, wood burning is worsening pollution in capitals such as Paris, Berlin and Lisbon.
Cities are moving toward it:
The mayor of London, Sadiq Khan, asked the Department for Environment, Food and Rural Affairs (Defra) in September for the power to ban solid fuel use (wood, coal and the like) in high-pollution areas of the city, starting in 2025, which would require amending the Clean Air Act.
If you live equatorward of 60°, there is in theory no need to actively heat your home. The most sustainable thing you can do, if you can afford it, is to upgrade your home to passive house standards and you can dispense with any sources of space heating altogether (you still need hot water, of course, but most wood fireplaces are impractical for this purpose).
There may exist specialised high pressure/temperature or perhaps heavily filtered wood fireplaces that exhaust much less or no particulate matter. If you insist on burning wood, investigate the particulate matter exhaust of your technological choice. Being carbon neutral is not enough; air quality must be part of your sustainability budget, in particular if you live in a populated area.
NB: This answer is focussing purely on what is best from a sustainability point of view. Of course, many people around the world cannot afford any other source of cooking or heating than wood fires, just like they cannot afford access to clean water, access to sewage, or the only vehicle they can afford is a very dirty scooter. Sustainable solutions often cost more out of pocket, which puts policy makers in a very difficult dilemma. Humanity must all work together to improve the economic situation of rural and urban people around the world such that we can all sustainably stay warm and cook. However, if you are purely asking from a sustainability point of view, there are far better alternatives than burning wood — in particular if your fireplace is typical.