Grep lines starting with 1, but not 10, 11, 100 etc

The question in the body

Select lines that start with a 1 and are followed by an space

grep -c '^1\s'          file
grep -c '^1[[:space:]]' file

That will also give the count of lines (without needing the call to wc)

The question in the title

A 1 not followed by another number (or nothing):

grep -cE '^1([^0-9]|$)' file 

But both solutions above have some interesting issues, keep reading.

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In the body of the question the user claim that the file is "tab delimited".

Delimiter

tab

A line starting with a 1 followed by a tab (an actual tab in the command). This fails if the delimiter is an space (or any other, or none):

grep '^1    ' file

space

A line starting with a 1 followed by a space (an actual space in the command). This fails if the delimiter is any other or none.:

grep '^1 ' file

tab or space

grep '^1(   | )' file
grep '^1[[:blank:]]' file

whitespace

A more flexible option is to include several space (horizontal and vertical) characters. The [:space:] character class set is composed of (space), \t (horizontal tab), \r (carriage return),\n(newline), \v (vertical tab) and \f (form feed). But grep can not match a newline (it is an internal limitation that could only be avoided with the -z option). It is possible to use it as a description on the delimiter. It is also possible, and shorter, to use the GNU available shorthand of \s:

grep -c '^1[[:space:]]` file
grep -c '^1\s'          file

But this option will fail if the delimiter is something like a colon : or any other punctuation character (or any letter).

Boundary

Or, we can use the transition from a digit to a "not a digit" boundary, well, actually "a character not in [_[:alnum:]] (_a-zA-Z0-9)":

grep -c  '^1\b' file       # portable but not POSIX.
grep -c  '^1\>' file       # portable but not POSIX.
grep -wc '^1'   file       # portable but not POSIX.
grep -c  '^1\W' file       # portable but not POSIX (not match only a `1`) (not underscore in BSD).

This will accept as valid lines that start with a 1 and are followed by some punctuation character.

Sounds like you just want this:

$ grep '^1\b' a
1        TGCAG.....
1        TGCAG......

For the counting portion of this:

$ grep -c '^1\b' file
2

With awk:

awk '$1 == "1" { print; x++ } END { print x, "total matches" }' inputfile