I say the breviary most of the time on my iPhone. It goes with me everywhere, and thus is extremely convenient; another advantage is the lack of flipping around of ribbons and pages.
I thought I might review a few breviary apps and sites, in case anyone else might care to try them out.
Universalis (and RC Calendar) £14.99 (Calendar is free)
If you say the Liturgy of the Hours in English, then this is probably the best available resource. It may seem expensive, but this is to pay off The Rapacious Grail for permission to use the official version of the psalms, and perhaps for the use of the Jerusalem Bible which is used for Scripture throughout. There is an option to use Universalis' own translation of the psalms (and in fact this is the default version). The Antiphons are not in the official translation in the Breviary, but are better (if a little too literal). Universalis (but not Calendar) also has hymns, real office hymns for the most part (a welcome change from the official Breviary). Universalis is entirely downloaded to your iPhone or iPad, which means that you can use it even if you do not have an internet connection. RC Calendar will download the office for the day, and will permit you to download up to ten days in advance (should one be going abroad, for instance), but no more. I don't think it uses the official Grail psalms, which might deter some who are obliged to the Office. You will also find the Mass readings for the day. Finally, Universalis has a number of diocesan calendars and propers also (though alas not my own).
If you want to say the Liturgy of the Hours in Latin, this is your only option. It seems also to have some sort of approbation from the Holy See. The texts are official, though the English version is according to the US breviary (fine if you're a North American!).
One peculiarity is attaching the Invitatory to Lauds: Universalis' solution of treating it as a separate item is much better, which gives one the option of celebrating the Office of Readings in the traditional place accorded to Matins.
iBreviary has been a little shaky in the past (as Fr Z observes), but they have recently revised it, and now it seems much better. I hear that they are planning to introduce an (altar) missal for the iPad soon. I'm not sure what I think about that.
Divinum Officium is not an app, but a web site, so you need to access it via Safari or another browser. It is a positively encyclopædic tool for celebrating the traditional office in either Latin or English; it meticulously presents the material in double columns throughout, and one column neatly fits into the iPhone's window and is very legible. You can access the office in virtually every shade of traditional Latin office, updated every day, from pure 1570 right up to 1961 with New Calendar (my own preference). It is a work of real skill, let down by mistakes from time to time (sometimes I have to reach for my printed breviary), and quite a lot of typos at Matins. Being a web site, internet connection is essential, at least to download each office.
A confusing name (I don't know which came first), this is also a web site. It is easier to read than Divinum Officium, having a grey/white background and a clearer, sans-serif, font, but is only in the 1961 version, and completely lacks Matins.