score:4

Accepted answer

You need to overwrite drilldown function, to avoid add action to labels.

http://jsfiddle.net/phpdeveloperrahul/FW64T/

 (function (H) {
    H.wrap(H.Point.prototype, 'init', function (proceed, series, options, x) {
        var point = proceed.call(this, series, options, x),
            chart = series.chart,
            tick = series.xAxis && series.xAxis.ticks[x],
            tickLabel = tick && tick.label;

        if (point.drilldown) {

            // Add the click event to the point label
            H.addEvent(point, 'click', function () {
                point.doDrilldown();
            });

            // Make axis labels clickable
            if (tickLabel) {
                if (!tickLabel._basicStyle) {
                    tickLabel._basicStyle = tickLabel.element.getAttribute('style');
                }
                tickLabel.addClass('highcharts-drilldown-axis-label')          .css({
                    'text-decoration': 'none',
                    'font-weight': 'normal',
                    'cursor': 'auto'
                    })
                    .on('click', function () {
                    if (point.doDrilldown) {
                        return false;
                    }
                });

            }
        } else if (tickLabel && tickLabel._basicStyle) {
        }

        return point;
    });
})(Highcharts);

score:0

In drilldown just put 

activeAxisLabelStyle: {
    cursor: 'pointer',
    color: '#0d233a',
    fontWeight: 'bold',
    textDecoration: 'none'          
}

score:0

In case you have a chart where only a selection of columns have drilldowns, Sebastian Bochan's answer needs to be modified so that all columns have the same label:

(function (H) {
            //DATALABELS
            H.wrap(H.Series.prototype, 'drawDataLabels', function (proceed) {
                var css = this.chart.options.drilldown.activeDataLabelStyle;
                proceed.call(this);

                css.textDecoration = 'none';
                css.fontWeight = 'normal';
                css.cursor = 'default';
                css.color = 'blue';

                H.each(this.points, function (point) {

                    if (point.dataLabel) { // <-- remove 'point.drilldown &&' 
                        point.dataLabel
                            .css(css)
                            .on('click', function () {
                                return false;
                            });
                    }
                });
            });
        })(Highcharts);

Also note that these settings are global, so also affect any other charts you may have.

score:1 

{...
series: [],
drilldown: {
    activeAxisLabelStyle: {
        cursor: 'default',
        color: '#3E576F',
        fontWeight: 'normal',
        textDecoration: 'none'
    },
    activeDataLabelStyle: {
        cursor: 'default',
        color: '#3E576F',
        fontWeight: 'normal',
        textDecoration: 'none'
    }
}
}

score:3

Or you can do it with css

.highcharts-drilldown-axis-label{
text-decoration: none !important;
}

score:5

We can use drilldown option to control the drilldown.

 drilldown: {
//for axis label
        activeAxisLabelStyle: {
            textDecoration: 'none',
            fontStyle: 'italic'
        },
//for datalabel
        activeDataLabelStyle: {
            textDecoration: 'none',
            fontStyle: 'italic'
        }
}

[Reference:http://jsfiddle.net/gh/get/library/pure/highcharts/highcharts/tree/master/samples/highcharts/drilldown/labels/][1]


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