Types of Accounts 

Offshore bank accounts are usually different from regular banking accounts.  They rarely offer an option to be able to write cheques.  In an onshore bank usually you choose between a chequing (checking) account and a savings account.  Onshore it doesn't matter if you bank at a trust company or a building society, both types of accounts are available.

Interest Rates in onshore savings & checking accounts vs Investment Accounts

The rate of interest paid offshore is often nothing or a very low interest rate.  The offsetting factor is that offshore accounts protect you from lawsuits and other risks to your assets.  Another possibilty to consider is an offshore investment account.  Be aware that banking and Investments accounts are very different.  Investment accounts allow you to trade stocks, options, forex (foreign exchange of currencies) but often don't allow you other amenities like sending a bank wire, online banking, and the like.  One possibility to overcome this shortcoming is to get your investment account at the same banking institution as your regular offshore bank account.

 Swiss Bank Account

If you bank in Switzerland, you have all the security of swiss banking, but all the requirements too.  Often you will need an introduction and a very large initial deposit.  The interest rates are competetive, but the Swiss government and financial institutions play ball with other countries including tax evasion issues.

 Panama Bank Account

Interestingly enough, Panama does not consider tax evasion a crime and as such will not co-operate with foreign governments.  When combined with a bearer share corporation, Panama is considered by many to be the "New Switzerland".  Essentially, unless you are a major criminal your banking secrets will be protected.  Panama's banking secrecy laws are second to none.  For that reason Panama is now the hottest offshore banking jurisdiction and tax haven. 

Building Society / Credit Union

Many quasi-banking institutions offer great rates onshore.  These options should be explored for all funds that you can afford to loose if you are sued.  Surplus funds beyond this amount should be kept in an offshore bank account. 

Trust Company 

A trust company is not a bank.  It is really a quasi-bank as mentioned above.  The same rules would apply to trust companies and building societies and credit unions.