Inkscape renders solid white as the transparent background by default, try switching to checkerboard background in the document properties for a second check on the white fills (Shift+Ctrl+D).
I don't think that's technically correct..... Or it might be correct, but reads in a way that isn't clear. Why would switching to the checkerboard background help? Because isn't the checkerboard visible even at 50%?
The Inkscape background, which shows up as white, by default, is in fact, transparent. Unless you changed it, it likely is still transparent. As far as I know, switching to the checkerboard background has no effect. The checkerboard background was added as an option in Inkscape, because most other graphics programs use it, to indicate transparency. And a lot of users like it.
Personally, the checkerboard bg drives me crazy. For objects with very low opacity, I can't see them against the checkerboard. For those tiny white areas (at the tip), I'm not sure if they would show up against the checkerboard.
But in any case, to make sure your background is transparent, you need to open Document Properties > Page tab. Then around halfway down, look at the Background section, and click on the color bar. If the canvas background is opaque, the bar will be a solid color. If it has any transparency, you'll see half of it is checkerboard. But since you need to make sure it's 100% transparent, you need to click on the bar, which will open a small color dialog. Make sure the A: slider (alpha) is all the way to the left, and showing a value of zero.
I can see some white areas - some tiny, some larger. But we would need to see the SVG file to tell for sure where they are coming from. I'd say Lazur's implication is correct, that at least some of the white areas are coming from a white fill on the paths. But those tiny white marks are the tip seem curious to me. So the SVG file can tell us for sure.
