Q:

Should contractions be avoided in formal emails?

In a formal email of the kind where you begin with "Dear Mr. Surname" and finish with "Best regards", for example, should we use the following contractions? Or are the non contracted forms more appropriate?

We have -> We've
We would -> We'd
There is -> There's
etc.

A:

In formal writing, do not use contractions.

A:

Giving blanket advice about contractions in "formal" writing is impossible because norms differ by discipline or purpose. When using contractions doesn't breach the specific governing norms, you should pursue the greatest clarity and concision, achieved by avoiding expanded verbs that are ordinarily contracted. I discuss this in depth in "The celebration of informality and the unsettled status of contractions." 

A:

Short answer: use contractions where appropriate.
The idea behind banning contractions is to avoid upsetting people who entertain the delusion that there’s something wrong with using them in a formal context. Before you decide that you don’t want to risk offending such people (in case there are any still living), it is important to beware the dangers of avoiding contractions:
You could sound pompous.
You risk trying the patience of those who don’t have a problem with contractions (i.e. anyone who reads your e-mail).
Care must be taken when rewording sentences so that meaning is preserved. For example these two sentences do not mean the same thing:

He thought a Christian could not attend church and still be saved.
He thought a Christian couldn't attend church and still be saved.

