We are not ruled by Brussels: on all the major areas of law - education, transport, health, taxation, benefits - our government has far greater influence than the EU.
And the EU is not unaccountable: it's accountable to directly elected representatives in the EU parliament, and to the democratically elected governments of the the member states.
The EU is nigh on unaccountable. MEPs can't draft laws, or amendments, only the commission can.
The commission isn't elected by the public, but instead chosen by the EU president, which is in turn elected by the MEPs from a list of candidates chosen by the council, which is made up of the heads of member states. Its democracy in the loosest possible sense of the word.
To decide have a say in the laws of the EU, we must elect a national government that agrees with us, hope that they can win the diplomatic contest to choose presidential candidates to out liking. Then vote for MEPs to agree this choice - under a vastly different voting system to the national elections - get the MEPs to form a large voting block with other MEPs by making policy compromises with other nations, and then hope that their vote is enough to secure the president of our choosing.
We must then hope that this new president, will elect members of the commission that we hope the UK agrees with (one per member state). Heads of member states have some say in this, but very little.
So then, having elected a capable head of state, who has won a 28 way battle to get our candidate of choice, and our MEPs voted on the winning side in the election, and the commission members chosen, we must then as a country, or special interest groups, or corporations, lobby the commision to draft laws and amendments to our liking. These are then passed to the MEPs to debate and vote upon - but not change directly - and we must hope that our MEPs can gather enough political influence to get a majority vote.
We've had one EU commission president, who was the choice of the labour party in the year that Thatcher was elected, and whose legacy was the EU monetary policy, which was an utter disaster for the UK.
Democratic and accountable eh?
Oh, and there is no appetite for reform of this system at all.
Lets then look at those listed major areas of law:
Education: Vast majority British. There is the €14B Erasmus programme, which funds students and lecturers to study in different institutes across the EU, common targets set for school leavers in all states, some incentives for members states to adopt specific policies, and the common recognition of all diplomas across the EU. There are longer term ambitions for greater powers, but they are very much low priority.
Transport: EU has domain over most elements of Air travel, haulage, shipping and road safety.
On rail they've made it so that any EU organisation can compete for any rail franchise across the EU (which is why most of our rail network is operated by France and Germany). Which makes it next to impossible to privatise the UK rail industry.
They also direct most rail infrastructure investment across the continent. They are one of the major driving forces for the construction of HS2, and are the reason why rail electrification and investment in the UK is directed north to south, rather than east to west.
http://ec.europa.eu/transport/themes/infrastructure/news/2015-05-28-coordinator-work-plans_en.htm
Health: Like education, the majority of health policy is decided in the UK. There is the EHIC system, and calls for health policies across the EU to be harmonised and opened up for competition, but again is very far into the future.
Taxation: Mostly British. VAT is harmonised across the EU, and the UK faces great difficulty in changing VAT laws. There are strong calls for harmonisation of corporation tax, as well as any taxes that might impact on the competition of labour between states.
The commission also advises all members states on their taxation, so that countries do develop vastly different systems. Long term you can expect powers to be increasingly decided by the commision.
In the case of a failing economy such as Greece, the EU steps in and decides everything.
Benefits: Again, mostly British. The little say that the EU does have over benefits in the UK was one of Cameron's main aims in the "renegotiation", and we all know how difficult it was for him to get his minor changes.
Here are some other areas of law.
Trade: The UK has next to no say whatsoever.
Immigration: Free movement of EU citizens is here to stay. We're not in Schengen, but we don't have full control over borders.
Business regulations: The EU decides the vast majority of this. Its widely considered to be very harmful to small firms in the UK.
Defence: Member states contribute to a small EU military force that mainly works with the UN. Calls for a greater EU army are being voiced by many within the EU in response the the Ukrainian conflicts, and work has begun in expanding the EU's military capability and its legal power to use it.
Crime: Mostly British, but europol and the EU arrest warrant means that this is starting to fall under EU control as well.
Foreign Policy: Harmonisation of Foreign policy and removing member state representation at international bodies such as the WTO and UN is one of the main objectives of the 5 presidents report, and will be enshrined into Law at the next treaty change.
Finance: The 5 presidents report looks to take control of the vast majority of financial regulation of the eurozone, and is will also have heavy implications on the city of London, though exactly how is unknown.