The Grammar Girl has a good article on this topic, basically: It is fine to use however at the beginning of a sentence; you just need to know when to use a comma.
If it means " to whatever extent ", don't use a comma:
However wrong it is, I will say it loud and clearly.
If it means " nevertheless ", use a comma:
However, I don't give a damn.
english.stackexchange.com/users/30/Edward Tanguay
I think this advice comes from the (somewhat strange) idea that sentences should have one complete idea.
If your sentence begins with the conjunction "however", then it's an extension of the idea in the previous sentence and is therefore not a "complete idea".
The same reasoning is behind advice not to begin sentences with "or" and "and".
Of course this advice is silly because sentences and "complete ideas" don't always line up, and there's no reason that a conjunction can't join a sentence with the previous one.
english.stackexchange.com/users/39/nohat
There is no rational, linguistically valid reason for this rule.
It's just a convention promulgated by misguided English writing teachers.
english.stackexchange.com/users/78/Alan Hogue
Yes.
However, it should be done sparingly.
Here's a pretty good writeup: http://grammar.quickanddirtytips.com/starting-a-sentence-with-however.aspx
english.stackexchange.com/users/8258/Eric King
However you parse the rule, this usage is correct.
However, this may not be.
I think both are fine, but they are different cases.
english.stackexchange.com/users/1696/TRiG
Yes, but you cannot start a story with it.
I also believe that there should be no more than one 'however' per three sentences.
english.stackexchange.com/users/188/Pavel Radzivilovsky
It is acceptable to start a sentence with however .
However, according to Strunk & White’s Elements of Style , it may only be used at the beginning of a sentence when it means “in whatever way” or “to whatever extent”.
When the word is used to mean “nevertheless”, it may not come first in its sentence or clause.
Strunk & White’s Misused Expressions
english.stackexchange.com/users/24351/youngburnside
No grounds whatsoever .
However, a teacher is a teacher , so if you want to get a higher grade, I would suggest "basking in your own glory" without telling your teacher they are in the wrong.
english.stackexchange.com/users/None/
I have heard that starting a sentence with  however  is wrong. What are the grounds for this view and is it still held by a majority of pedants?  They would suggest changing   However, some people are beginning to doubt this.   to   Some people, however, are …   or   Some people are, however, beginning …
Is it acceptable to start a sentence with “however”?
english.stackexchange.com
<grammaticality><word-order><conjunctions><sentence-starts><discourse-markers>
english.stackexchange.com/questions/79
english.stackexchange.com/users/60/bryn
It is generally agreed that it is perfectly acceptable to use "however" at the beginning of a sentence. There are people who claim that there is no rational reason for such a rule preventing its use. It is fine as long as you use a comma when it means 'nevertheless' and no comma when it means 'to whatever extent'.
It is generally agreed that it is perfectly acceptable to use "however" at the beginning of a sentence. There are people who claim that there is no rational reason for such a rule preventing its use.  It is acceptable to start a sentence with  however, but you cannot start a story with it.  According to Strunk & White’s  Elements of Style , it may only be used at the beginning of a sentence when it means “in whatever way” or “to whatever extent”.  When the word is used to mean “nevertheless”, it may not come first in its sentence or clause.  
It is acceptable to use "however" at the  beginning of a sentence, but you need  to know when to use a comma. There are no rational grounds for this rule and it is perfectly acceptable to start a sentence with the word "however".
It is fine as long as you use a comma when it means 'nevertheless' and no comma when it means 'to whatever extent'. The advice is based on the incorrect idea that a sentence should have one complete idea, and can therefore be ignored. This is a misguided convention. It's OK if done sparingly. Both are correct in different cases. This is a pedantic rule.
 It is acceptable to start a sentence with  however, but you cannot start a story with it.   According to Strunk & White’s  Elements of Style , it may only be used at the beginning of a sentence when it means “in whatever way” or “to whatever extent”.  When the word is used to mean “nevertheless”, it may not come first in its sentence or clause.  
Yes, it is perfectly acceptable to start a sentence with the word "however". There are no rational grounds for this rule.