Sorry just re-read your first post, I thought you were just talking about dividing up the house contents at first glance for some reason yesterday!
The proceeds of the house should be discussed alongside the broader financial settlement between you and your ex. So you'll both need to draw up what assets you have (house, savings/investments, pension) and your liabilities (loans, mortgage, credit cards, etc), and then negotiate who gets what.
The starting point for discussion will be a 50:50 split across all assets, but it doesn't often ultimately work out that way, as the split of assets will differ according to "needs". Typically the person who a) has the children the most and b) earns the least, especially if they have sacrificed a career to look after children will get a larger share of the assets.
Often what can happen is that one party can trade off their share of a pension, for example, to get a higher proportion of the house - this can make it much easier to afford to buy one or even stay in the existing house, but it will depend on your individual circumstances and preference. You should be able to get a good steer from your solicitor as to your own circumstances.
As I said in my first post, who keeps the actually contents of the house should be negotiated, but I would certainly recommend not bickering over the small stuff if possible.