One of the reasons that you might wish to pursue such a path is because it may help to put more pressure on your ISP and encourage them to solve the problem, especially if there's an implied threat of bad publicity involved. Please see below for a list and explanation of other helpful organisations and services.
This not-for-profit professional membership association was setup to help protect and advise on consumer rights in relation to illegal sales activity. It deals with a range of issues from rogue tradesmen to under age knife sales and was used a lot against bad ISPs prior to the inception of Ofcom's official ADR complaints handler schemes (it's now best to use the ADR process).
Trading Standards is most likely to be of use when your complaint relates to having purchased a service where some or all aspects have not been delivered (miss-sold) or perhaps even seriously misrepresented (remember that broadband is a "best efforts" service, so issues such as poor speeds are not really valid for this organisation).
Note: Sadly Trading Standards cannot help with individual complaints directly, but they do work with the Citizens Advice Bureau (no.3 below) to tackle such issues and some of their local offices may be able to help if the ISP you're having difficulty with is also based locally.
Trading Standards Website
http://www.tradingstandards.gov.uk
https://www.gov.uk/find-local-trading-standards-office
The CMA was perhaps better known by its former title of the Office of Fair Trading (OFT), which came to an end on 1st April 2014. The new CMA is a non-ministerial department of Government that works to promote competition for the benefit of consumers, both within and outside the United Kingdom.
As before the CMA focuses more on larger scale problems than individual complaints and seeks to ensure a level playing field for various markets. It has the power to make businesses comply with UK competition and consumer laws and to improve trading practices.
Generally the OFT is best used as an information source, although if you and many others make a complaint to Ofcom, the CAB (no.3 below), business groups or trade bodies (e.g. ISPA or the Federation of Small Businesses etc.) then extreme situations may end up involving the CMA.
CMA Website
http://www.gov.uk/cma
The Citizens Advice Bureau (CAB) was setup to help people resolve their legal, money and other problems by providing free, independent and confidential advice; as well as by influencing policymakers. This is an especially good way of gaining free legal advice about consumer issues, particularly if you're considering any kind of court action against an ISP.
The CAB can also assist with making a complaint to Trading Standards and they will pass it to the appropriate local authority centric trading standards service. You can either do this via the CAB's website or report a complaint via their helpline on 03454 04 05 06. However if your complaint relates to serious fraudulent activity by an ISP then inform Action Fraud UK (note: minor billing/admin errors don't really count).
CAB Website
http://www.adviceguide.org.uk
The most serious ISP disputes usually concern a mix of money and contractual obligations. These often involve smaller amounts of cash and thus make big court cases pointless. Instead, provided you have clear reason and evidence to prove your case, then a request for compensation can be made via the Small Claims Court. You can start a small claim via any court in England or Wales (note: Scotland adopts a slightly different approach).
Typically such action, which will be handled by your local county court system, should only be taken as a LAST RESORT and after the OFFICIAL ADR SCHEME has proven itself to be ineffective. Most cases will be fairly simple, although very complicated cases or those seeking to claim more than £10,000 might be referred to a higher more costly authority.
As you'd expect there is a fee for using the small claims court, which is calculated based on the amount you are claiming, including interest. The fee (usually £25 or more) is not charged for those on income support, State Pension Guarantee Credit, income-based Jobseeker's Allowance, Working Tax Credit with no element of the Child Tax Credit or if you have a tiny income.
Small Claims Court (Information and Fees)
https://www.gov.uk/make-court-claim-for-money/overview
Small Claims Court (Scotland)
http://www.scotcourts.gov.uk/taking-action/small-claims
This refers to websites like ours - ISPreview.co.uk, which can help to publicise problems that ISP users might be experiencing. In order to do this we require that readers inform us about their problems because most of the time we can only cover widespread issues and we won't know if they're widespread unless you tell us about them (catch 22).
At this point readers can either choose to email the editor (Mark Jackson) via our Contact page or post a message on our forum (ISP Complaints Forum - requires registration to post).
Similarly we run a Reader ISP Reviews system which, once registered via our forum, will allow you to post and rate your ISP. Informing others about your experiences, be they positive or negative, is a very powerful tool indeed. Help us to help you (Tip: keep your reviews short, fair and to the point; people rarely read long-winded posts).