At what temperatures can you get frost?

There are a huge amount of variables with regards to whether you will get frost or not and going by what you have said, I'm assuming that you are talking about ground frost, not air frost.
Temperature would of course be the most important thing. The saturation of the air would probably be next followed by wind strength however if the air temperature is below 0c, wind is slightly less important. Whether or not there is an inversion in place can also play a role.
Ground (and air temperatures) can vary widely over short distances. Cold air sinks and it tends to look for the lowest point. if you you are in a rural area and you have a bowl shaped area that is lower than surrounding areas, then on a calm night, the bowl area is likely to be colder than surrounding areas.

For example, the weather station at Dublin Airport is in a known frost hollow and on calm cool nights, it often experiences very low temperatures in comparison to other weather stations in the area, despite being fairly close to the sea. Air temperatures got down to -1c last night and ground temperatures were probably several degrees lower than that. Casement, at the foothills of the Dublin/Wicklow mountains only got to +3C, despite being further from the sea.
The above are official weather stations. Other unofficial stations near Dublin airport (probably in suburban gardens) had the following minimums last night:
Artane - 4.1c
Glasnevin - 2c
Baldoyle - 4.6c
Swords West - 4.1c
Me (North kildare, near Enfield) - 2.6c

it's quite possible that you live in a frost hollow. Without giving exact details of your address, if you could give more exact details of where you are, I can check out your topography.....PM if you like if you don't want to give out those details on the public forum.

All the temperatures were + except for Dublin Airport. Apologies for the confusion with you thinking they were all minus's.....I totally see why you would have thought that.

A low of 2c can very easily give a grass/ground frost. Last week I had to scrape the frost off my car 4 early mornings out of 5. None of the mornings had an air temperature below 0c and one of the frosty mornings had an overnight air temperature minimum of 3.5c. The reason why you can get a ground frost with air temperatures above 0c is because the ground radiates heat very quickly once solar heating is not present. Some materials radiate heat more quickly than others. The 2 main ones I can think of is the metal surfaces of cars and slate rooves. These lose heat more quickly than the surrounding air can cool so they reach 0c before the surrounding air does so.

Where I live, I am situated near the bottom of a small valley. On 3 of the 4 frosty mornings last week, there was frost in my garden and in the surrounding fields but only up to about 15/20m above the lowest point of the valley. This was because the cold air pooled down to the lowest point as the night went on. There was no wind present so the cold air built up to a point that frost could develop in the grass. The higher up the hill I went, the less cold I got.

As for your situation last night, the winds were very light during the night but what little wind was there was generally from a direction between north and north east. Saint Annes Park is to your northeast and this is a large area of what is essentially rural grassland. It is also situated at a slightly higher altitude to your home. It's very much possible that the cold air that built up in the park slowly meandered it's way with the prevailing wind and also downhill to you, giving you lower than expected temperatures. I would say your ground temperature barely got below freezing and air temps were probably a couple of degrees above freezing.

The Phoenix park is a great example of how a large rural type area on a slope can have several different weather types. During winter, it is not at all uncommon to enter the park at the Castleknock entrance and be in bright sunshine with clear skies and temperatures of 4 or 5 degrees (or higher). Head down Chesterfield Avenue and exit the Park at Islandbridge and you may well be in dense freezing fog with a white frosty landscape all round. This is because the cold air in the park rolls downhill into the Liffey valley. It builds and builds all night and eventually an inversion sets up over the valley - the cold air sits at the bottom of the Liffey valley and as it is so enclosed, it can't escape. Meanwhile, the air immediately above is maybe 5 or 6c warmer. if the humidity is high enough, the air becomes so saturated that it can not remain as a gas any more and it condenses, forming fog. If the temperature gets below freezing, it becomes freezing fog.

Back to you in Clontarf last night, had the wind been from NE clockwise through to south, then you would have had a much "warmer" sea breeze. The sea would have kept the temperatures much higher (humidity would have been higher too as the air would have passed over a large volume of relatively warm water) and you would not have had frost.

That actually would be very interesting to hear back of any findings you do get over time. Obviously, what I said may not be true at all. I was just looking at he geography and topography of the area combined with the weather last night and made an educated suggestion as to what I personally thought what may have been the reason for your low temps.
Originally not too far from the you so am familiar enough with the area.