I think you are misunderstanding. ALL analog scopes ALWAYS show a single dot. The phosphor that glows to produce the display has a decay time, it doesn't extingush imediately so if the sweep time is fast enough, it replenishes the glow before it has had time to decay and hence it looks like a line rather than a dot is being shown. An analog storage scope and all digital scopes work on a slightly different principle. Instead of displaying the waveform 'live', they repeatedly measure the signal voltage at short intervals using an ADC and store the result in a memory. The values represent the shape of the waveform, rather like the Y axis values if you plotted the waveform versus time on a graph. The X axis values are the sample number stored in the memory. By reading the samples at higher speed and moving the dot along the X asis as you do so, then moving the dot up or down according to the sample value, you redraw the digitized waveform. The big advantage of this is you can 'loop' the reading of the memory at a higher speed than you made the recording so the dot redraws the screen fast enough that the waveform looks continuous.
The kind of waveform and the sweep speed are not related, an oscilloscope display is simply a graph of voltage (your waveform) against time (the sweep speed). You adjust the sweep speed so you can observe the relevant part of the waveform you want.
In single sweep mode, you will never 'lock' the waveform. Single sweep does just that, you have to press the reset button before it will sweep again. However, it can be very useful to find the best trigger level setting. If you select single sweep mode and keep pressing reset while adjusting the trigger level, you should find a level that waits for the start of your waveform before the sweep starts. Leave the trigger level at that setting and the go back to continuous sweep mode, it should now lock the waveform to the display.
As far as setting the optimum sweep speed, consider what you want to measure. It's a time interval. The X axis is calibrated in small divisions across the graticule (graph lines on the display), how long each of these represents will depend on the sweep speed you set using the timebase control. You will see it is marked something like "Secs/Division", each 'division' is one square on the graticule. So if you selected 1 second/division, the dot would travel across one division in one second. Using that setting, your .28 second waveform would last .28 divisions. If you increased the timebase speed to say 0.1s/division, the dot would travel ten times faster and your waveform would last 2.8 divisions on the graticule. You select the sweep speed that best scales your waveform so you can count the graticule divisions easiest. The time will always be the number of divisions multiplied by the time it takes the dot to travel per division, in other words 'sweep speed x length of waveform'.
Brian.