Pattern matching with LIKE, SIMILAR TO or regular expressions in PostgreSQL I had to write a simple query where I go looking for people's name that start with a B or a D : 

SELECT s.name 
FROM spelers s 
WHERE s.name LIKE 'B%' OR s.name LIKE 'D%'
ORDER BY 1


I was wondering if there is a way to rewrite this to become more performant. So I can avoid or and / or like?
Your query is pretty much the optimum. Syntax won't get much shorter, query won't get much faster:
SELECT name
FROM   spelers
WHERE  name LIKE 'B%' OR name LIKE 'D%'
ORDER  BY 1;

If you really want to shorten the syntax, use a regular expression with branches:
...
WHERE  name ~ '^(B|D).*'

Or slightly faster, with a character class:
...
WHERE  name ~ '^[BD].*'

A quick test without index yields faster results than for SIMILAR TO in either case for me.
With an appropriate B-Tree index in place, LIKE wins this race by orders of magnitude.
Read the basics about pattern matching in the manual.
Index for superior performance
If you are concerned with performance, create an index like this for bigger tables:
CREATE INDEX spelers_name_special_idx ON spelers (name text_pattern_ops);

Makes this kind of query faster by orders of magnitude. Special considerations apply for locale-specific sort order. Read more about operator classes in the manual. If you are using the standard &quot;C&quot; locale (most people don't), a plain index (with default operator class) will do.
Such an index is only good for left-anchored patterns (matching from the start of the string).
SIMILAR TO or regular expressions with basic left-anchored expressions can use this index, too. But not with branches (B|D) or character classes [BD] (at least in my tests on PostgreSQL 9.0).
Trigram matches or text search use special GIN or GiST indexes.
Overview of pattern matching operators

LIKE (~~) is simple and fast but limited in its capabilities.
ILIKE (~~*) the case insensitive variant.
pg_trgm extends index support for both.

~ (regular expression match) is powerful but more complex and may be slow for anything more than basic expressions.

SIMILAR TO is just pointless. A peculiar halfbreed of LIKE and regular expressions. I never use it. See below.

% is the &quot;similarity&quot; operator, provided by the additional module pg_trgm. See below.

@@ is the text search operator. See below.


pg_trgm - trigram matching
Beginning with PostgreSQL 9.1 you can facilitate the extension pg_trgm to provide index support for any LIKE / ILIKE pattern (and simple regexp patterns with ~) using a GIN or GiST index.
Details, example and links:

How is LIKE implemented?

pg_trgm also provides these operators:

% - the &quot;similarity&quot; operator
&lt;% (commutator: %&gt;) - the &quot;word_similarity&quot; operator in Postgres 9.6 or later
&lt;&lt;% (commutator: %&gt;&gt;) - the &quot;strict_word_similarity&quot; operator in Postgres 11 or later

Text search
Is a special type of pattern matching with separate infrastructure and index types. It uses dictionaries and stemming and is a great tool to find words in documents, especially for natural languages.
Prefix matching is also supported:

Get partial match from GIN indexed TSVECTOR column

As well as phrase search since Postgres 9.6:

How to search hyphenated words in PostgreSQL full text search?

Consider the introduction in the manual and the overview of operators and functions.
Additional tools for fuzzy string matching
The additional module fuzzystrmatch offers some more options, but performance is generally inferior to all of the above.
In particular, various implementations of the levenshtein() function may be instrumental.
Why are regular expressions (~) always faster than SIMILAR TO?
The answer is simple. SIMILAR TO expressions are rewritten into regular expressions internally. So, for every SIMILAR TO expression, there is at least one faster regular expression (that saves the overhead of rewriting the expression). There is no performance gain in using SIMILAR TO ever.
And simple expressions that can be done with LIKE (~~) are faster with LIKE anyway.
SIMILAR TO is only supported in PostgreSQL because it ended up in early drafts of the SQL standard. They still haven't gotten rid of it. But there are plans to remove it and include regexp matches instead - or so I heard.
EXPLAIN ANALYZE reveals it. Just try with any table yourself!
EXPLAIN ANALYZE SELECT * FROM spelers WHERE name SIMILAR TO 'B%';

Reveals:
...  
Seq Scan on spelers  (cost= ...  
  Filter: (name ~ '^(?:B.*)$'::text)

SIMILAR TO has been rewritten with a regular expression (~).
Ultimate performance for this particular case
But EXPLAIN ANALYZE reveals more. Try, with the afore-mentioned index in place:
EXPLAIN ANALYZE SELECT * FROM spelers WHERE name ~ '^B.*;

Reveals:
...
 -&gt;  Bitmap Heap Scan on spelers  (cost= ...
       Filter: (name ~ '^B.*'::text)
        -&gt;  Bitmap Index Scan on spelers_name_text_pattern_ops_idx (cost= ...
              Index Cond: ((prod ~&gt;=~ 'B'::text) AND (prod ~&lt;~ 'C'::text))

Internally, with an index that is not locale-aware (text_pattern_ops or using locale C) simple left-anchored expressions are rewritten with these text pattern operators: ~&gt;=~, ~&lt;=~, ~&gt;~, ~&lt;~. This is the case for ~, ~~ or SIMILAR TO alike.
The same is true for indexes on varchar types with varchar_pattern_ops or char with bpchar_pattern_ops.
So, applied to the original question, this is the fastest possible way:
SELECT name
FROM   spelers  
WHERE  name ~&gt;=~ 'B' AND name ~&lt;~ 'C'
    OR name ~&gt;=~ 'D' AND name ~&lt;~ 'E'
ORDER  BY 1;

Of course, if you should happen to search for adjacent initials, you can simplify further:
WHERE  name ~&gt;=~ 'B' AND name ~&lt;~ 'D'   -- strings starting with B or C

The gain over plain use of ~ or ~~ is tiny. If performance isn't your paramount requirement, you should just stick with the standard operators - arriving at what you already have in the question.