I am not sure whether this is a file location problem or if it is a problem of the svg you use. As long as it is within the root folder of your app I can not see why is not working. Dont you have any firebug errors?
try to use the following snip to check if it is working. I have tried it works fine.
var style = new ol.style.Style({
image: new ol.style.Icon({
src: 'http://www.williambuck.com/portals/0/Skins/WilliamBuck2014/images/location-icon.svg'
})
});
Also consider removing the size you provide within your code snip. So call it like that
var style = new ol.style.Style({
image: new ol.style.Icon({
src: "styles/transport_aerodrome.svg"
})
});
UPDATE
if you can get the xml out of your svg file. try the following:
var svg = '<svg version="1.1" id="Layer_1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" x="0px" y="0px" width="30px" height="30px" viewBox="0 0 30 30" enable-background="new 0 0 30 30" xml:space="preserve">'+
'<path fill="#156BB1" d="M22.906,10.438c0,4.367-6.281,14.312-7.906,17.031c-1.719-2.75-7.906-12.665-7.906-17.031S10.634,2.531,15,2.531S22.906,6.071,22.906,10.438z"/>'+
'<circle fill="#FFFFFF" cx="15" cy="10.677" r="3.291"/></svg>';
var mysvg = new Image();
mysvg.src = 'data:image/svg+xml,' + encodeURIComponent(svg);
//and then declare your style with img and imgSize
var style = new ol.style.Style({
image: new ol.style.Icon({
img: mysvg,
imgSize:[30,30]
})
});