You could calculate lat and lon as attributes in you shapefile and then use the "select features using an expression" tool. Important to make sure that the lat and lon are using the same precision and scale as the coordinate that you are searching for.
Amended - included steps for using the "Select features using an expression tool"
You mentioned in your comment that you have extracted nodes and then calculated the geometry. So I am assuming you point attribute table looks like mine below with a "XCOORD" and "YCOORD" column.
You then want to click on the "Select features using an expression tool button" shown below in the red circle.
This opens a dialog where you can create an expression using your attribute columns and values. For example below I am going to query the "XCOORD" attribute for all values that are equal to 146.587500, hit the select button at the bottom right to execute the expression.
The result will be a selection in your attribute table which you can export to a new layer if required. My resultant selection returned two points out of 147 with the same "XCOORD".
With the method make sure that the value you are searching for in the expression tool is an exact match to the value you want to find in your table.
Alternatively if the shapefile is of a large shoreline consider putting it in PostGIS. This could speed things up for you, but probably only worth doing if it something that you'll do more than once. If you have the nodes of the shoreline you could then do a query on the geometry for all values that equal a certain latitude. For example
SELECT * FROM table WHERE ST_Y(the_geom)::decimal(5,3) = -35.883